This North African species was described in 1870 based on fossil remains that were dated to the Upper Pleistocene, however, some remains apparently turned out to be much younger, thus the species might have survived until about 3000 years BP..
I was unfortunately unable to find any source for that assumption.
The Mt. Kahuzi Climbing Mouse was described in 1969; the species is known only from two specimens, a male and a female that were collected in 1967 and 1972 respectively.
The species is (or was) known only from the forests at Mount Kahuzi in the Democratic Republic of the Congo; it is highly threatened by illegal logging and habitat destruction and, having never been found since 1972, might actually already be extinct.
Radofilao’s Sloth Lemur was described in 1990 based on subfossil remains that were found in some limestone caves along the northern coast of Madagascar, and which could be dated to ages of about 3100 to 2800 BCE..
This species is known from subfossil remains that were recovered from archaeological sites on the small island of La Desirade off the northeast coast of Guadeloupe in the Lesser Antilles.
The remains were dated to about 600 to 1400 AD.. 
 S. T. Turvey: Holocene Extinctions. Oxford University Press, USA 2009
 Myriam Boudadi-Maligne; Salvador Bailon; Corentin Bochaton; Fabrice Casagrande; Sandrine Grouard; Nathalie Serrand; Arnaud Lenoble: Evidence for historical human-induced extinctions of vertebrate specieson La Désirade (French West Indies). Quaternary Research 85: 54-65. 2016
This species was described in 1899 based on subfossil remains that had been found in Madagascar.
The Large Sloth Lemur was apparently restricted to the southern part of Magadascar
The species is known to have survived into the 17th century, bones from a remote cave in the interiror of southern Madagascar could be dated to an age of about 1285 to 1625 AD. The species is mentioned in Malagasy folklore accounts as the tratratratra or tretretretre and was also mentioned by Étienne de Flacourt, who was the french governor of Madagascar from 1648 to 1655.:
“The Tretretretre, or Tratratratra, is an animal as large as a two-year- old calf, and which has a round head and a man’s face; the fore-feet like those of a Monkey, and the hind ones also. It has curled hair, a short tail, and ears like those of a man. … It has been seen near Lake Lipomani, in the vicinity of which is ils lair. It is a very solitary animal, which the people of the country hold in great fear and run away from, as it also does from them.” 
I have not yet read the original, which is in French, and there are several translations, which sometimes mention a Lake Lipomani or a Lipomani Lagoon, which I could not detect yet.
 Étienne de Flacourt: Histoire de la grande Isle Madagascar composée par le Sieur de Flacourt, Directeur Général de la Compagnie Françoise de l’Orient, & Commandant pour sa Majesté dans ladite Isle & és Isles adjacentes. Avec une Relation de ce qui s’est passé és année 1655. 1656. & 1657. non encore veuë par la premiere Impression, Paris: Gervais Clovzier 1661  David A. Burney; Ramilisonina: The Kilopilopitsofy, Kidoky, and Bokyboky: Accounts of strange animals from Belo-sur-mer, Madagascar, and the megafaunal “extinction window”. American Anthropologist 100(4): 957-966. 1999
This species was described in 2011 based on the dried skin and skull of an adult male that had been collected in 1937; it was apparently restricted to a small area at Mt. Choqa in north-western Ethiopia
It is believed that the species was last seen in 1968; it might now be extinct.
This species was described in 1887 on the basis of fossils that were found in Pleistocene deposits in Brazil; additional material was later also found in deposits that could be dated to the Early Holocene.
The same species, however, was recently recorded based on subfossil remains from the Late Holocene of Argentinia, thus it appears to have survived there and was then extirpated by humans, maybe even after the arrival of the first European settlers in the 15th century. 
 Diego H. Verzi; A. Itarí Olivares; Patricia Hadler; Juan C. Castro; Eduardo P. Tonni: Occurrence of Dicolpomys (Echimyidae) in the late Holocene of Argentinia: The most recently extinct South American caviomorph genus. Quaternary International 490: 123-131. 2018
The Mace-tailed Glyptodon was a remarkable giant armadillo that inhabited large parts of the South American continent.
The species reached a heigth of about 1,5 m and was up to 3,6 m long; its weight is estimated to have been about 1400 to 2370 kg.
This is very likely a Pleistocene species, that disappeared at the end of that era, however, there are some remains that had been dated to an age of 8000 to 7000 years ago, which suggests that it in fact might have survived into the earliest Holocene; these dates, however, are questioned by some scientists.
Pemberton’s Deermouse was restricted to the Isla San Pedro Nolasco in the Culf of California, Baja California, Mexico.
The species reached a legth of about 21 cm (including the tail), its fur was cinnamon-colored and flecked with some fine darker lines the head was slightly ligther colored and the belly was white.
Pemberton’s Deermouse was described in 1932 based on 12 specimens that had been collected on December 26, 1931 by Dr. William Hendy Burt of the California Institute of Technology, it was never found again.
The Estanque Deermouse was endemic to the Isla Estanque, a tiny, only 0,83 km² large islet south to Isla Ángel de la Guarda in the Gulf of California, Baja California, Mexico. However, this particular population apparently was never officially described and may in fact have been identical to the nominate subspecies inhabiting Isla Ángel de la Guarda.
Anyway, whether endemic or not, the deermice of Isla Estanque are gone now, the whole population appears to have been wiped out within a single year (1998 to 1999) by a single feral cat! 
 Ella Vázquez-Domínguez; Gerardo Ceballos; Juan Cruzado: Extirpation of an insular subspecies by a single introduced cat: the case of the endemic deer mouse Peromyscus guardia on Estanque Island, Mexico. Oryx 38(3): 347-350. 2004
The Hispaniolan Edible Rat was described in 1916 based on subfossil remains.
Gonzalo Fernández de Oviedo y Valdés, a Spanish colonialist, historian, and writer gave some brief descriptions of several mammal species that lived on the island of Hispaniola in the early 16th century; most are second-hand accounts of animals he himself had not seen; there is also one account that can be attributed to the Edible Rat.:
“The mohuy is an animal somewhat smaller than the hutia: its color is paler and likewise gray. This was the food most valued and esteemed by the caciques and chiefs of this island; and the character of the animal was much like the hutia except that the hair was denser and coarser (and more stiff), and very pointed and standing erect or straight above. I have not seen this animal, but there are many who declared it to be as aforesaid; and in this island there are many persons who have seen it and eaten it, and who praise this meat as better than all the others we have spoken about.” 
The species died out shortly after the arrival of the first European settlers in the Caribbean, who brought with them mice and rats, which very likely were the main reason for the extinction of most smaller endemic mammal species.
 Gerrit S. Miller: Mammals eaten by Indians, Owls, and Spaniards in the coast region of the Dominican Republic. Smithsonian Miscellaneous Collections 82(5): 1-16. 1929
Photo from: ‘Gerrit S. Miller: Mammals eaten by Indians, Owls, and Spaniards in the coast region of the Dominican Republic. Smithsonian Miscellaneous Collections 82(5): 1-16. 1929’
This species is known only from subfossil remains that were recovered from deposits from the Andrahomana cave in southeast Madagascar.
The remains can be dated to an age of about 3000 years.
The photo below shows another species from the same genus, the Short-tailed Shrew Tenrec (Microgale brevicaudata G. Grandidier) from northern and northwestern Madagascar.
 S. M. Goodman; N. Vasey; D. A. Burney: Description of a new species of subfossil shrew tenrec (Afrosoricida: Tenrecidae: Microgale) from cave deposits in southeastern Madagascar. Proceedings of the Biological Society of Washington 120(4): 367-376. 2007
The Alor Giant Rat is known from subfossil remains that were recovered from cave deposits on the island of Alor in the Lesser Sundas, Indonesia.
The species was larger than most recent rat species and might have survived into quite recent times, the authors of the description in fact assume that it might be still alive today. 
 Julien Louys; Sue O’Connor; Mahirta; Pennilyn Higgins; Stuart Hawkins; Tim Maloney: New genus and species of giant rat from Alor Island, Indonesia. Journal of Asia-Pacific Biodiversity 11(4): 503-510. 2018
Mejia Island Deermouse (Peromyscus guardia ssp. mejiae)
The Mejia Island Deermouse, described in 1932, was a subspecies of the Guarda Deermouse (Peromyscus guardia Townsend) and was endemic to the tiny Isla Mejía in the Gulf of California, Baja California, Mexico.
This mouse was last recorded in 1973 but disappeared sometimes after that date due to predation by introduced cats and competition by likewise introduced mice and rats. 
 Erik Mellink; Gerardo Ceballos; Jaime Luévano: Population demise and extinction threat of the Angel de la Guarda deer mouse (Peromyscus guardia) Biological Conservation 108: 107-111. 2007
This species is known from subfossil remains that were excavated on the island of Grenada in the Lesser Antilles.
The species was formerly just named as Cricetidae gen. & sp. ‘Grenada’, it was finally formally described in 2021.
 Brittany A. Mistretta; Christina M. Giovas; Marcelo Weksler; Samuel T. Turvey: Extinct insular oryzomyine rice rats (Rodentia: Sigmodontinae) from the Grenada Bank, southern Caribbean. Zootaxa 4951(3): 434-460. 2021
The Jamaican Rice Rat was described in 1898; it was restricted to the island of Jamaica.
The species reached a size of about 26 cm (including the tail); it was furthermore described as: “Color. Above rufous sparsely lined with black, brightest on the rump; head suffused with grayish; under parts yellowish, hairs gray at base; tail pale brown above, lighter beneath; hands and feet whitish; ears blackish outside, yellowish inside.“. 
In 1872, Small Indian Mongooses (Urva auropunctata (Hodgson)) had been imported to Jamaica to control the likewise introduced rats in the sugarcane plantations; the mongooses, however, were also very effective in eradicating the native rodents. Feral cats and dogs certainly played their part too and finally, in 1877, the Jamaican Rice Rats were seen for the last time.
 Daniel Giraud Elliot: The Land and Sea Mammals of Middle America and the West Indies. Publication of the field Columbian Museum, Zoological Series 4(1). 1904
Photo from: ”Clayton E. Ray: The Oryzomyine Rodents of the Antillean Subregion. Doctor of Philosophy thesis, Harvard University, 2111 pp. 1962′
Darwin’s Galapagos Mouse was described in 1929; it was endemic to the Isla Santa Cruz in the Galápagos archipelago.
It reached lengths of about 22 cm, including the tail; it was predominantly cinnamon rufous colored, the upper parts with a mixture of blackish hairs, the tail was dusky above and whitish below. 
Darwin’s Galapagos Mouse was last seen in 1930; the species disappeared shortly after, perhaps due to the introduction of Norway Rats (Rattus norvegicus (Berkenhout)), Black Rats (Rattus rattus (L.)), and feral cats.
 Wilfred H. Osgood: A new rodent from the Galapagos Islands. Field Museum of Natural History 17(2): 21-24. 1929
The Saba Rice Rat is known from subfossil remains that were recovered from an archaeological site named Kelbey’s Ridge on the small island of Saba in the Caribbean Netherlands.
The remains of this species were dated to about 1290 to 1400 AD.. 
 Elizabeth Reitz; C. Margaret Scarry; Sylvia J. Scudder: Case Studies in Environmental Archaeology (Interdisciplinary Contributions to Archaeology). Springer; Second Edition 2007  S. T. Turvey: Holocene Extinctions. Oxford University Press, USA 2009
Emma’s Giant Rat was described in 1994, based on a single female specimen that had been cought in 1964, it apparently was endemic to the very small island of Owi, near Biak in the New Guinean part of Indonesia.
The species is believed to have mainly been arboreal, nothing else is known about its biology.
The White-eared Pocket Mouse was described in 1894; the species, which is split into two subspecies, is apparently restricted to the San Bernardino- and Tehachapi Mountains in California, USA.
The species, together with two additional closely related ones, is considered a Pleistocene relict.
The nominate race was restricted to a small area of around 4 km² at Strawberry Peak and Squirrel Inn in the San Bernardino Mountains, where it was recorded between 1920 and 1933; it was not found since and is thus believed to be extinct. 
 Philip V. Brylski: White-eared Pocket Mouse, Perognathus alticola alticola. Terrestrial Mammal Species of Special Concern in California, Bolster, B. C., Ed., 1998. 102-104
The Giant Vampire Bat was described in 1988 based on bones that were recovered from deposits of a cave in the state of Monagas, northern Venezuela. Further remains were found in Argentina, Belize, Bolivia, Brazil, Ecuador, and Mexico.
The exact age of these bones cannot be determined, they may be of late Pleistocene or early Holocene age, some scientists even think that this species may still exist. 
The Giant Vampire Bat wasn’t a real giant, in fact it was only 30% larger than its next living relative, the Common Vampire Bat (Desmodus rotundus Geoffroy) (see photo).
 G. S. Morgan; O. J. Linares; C. E. Ray: New species of fossil vampire bats (Mammalia, Chiroptera, Desmodontidae) from Florida and Venezuela”. Proceedings of the Biological Society of Washington. 101(4): 912–928. 1988
The New Caledonian Roundleaf Bat is known from subfossil remains that were recovered from Holocene deposits in the Mé Auré Cave near the southern shore of Grande Terre, New Caledonia.
The remains were compared to the bones of several other congeneric species from the nearest regions but did not match to any of them, thus it most likely was a distinct species that might well have been endemic to New Caledonia. 
 Suzanne J. Hand; Jack A. Grant-Mackie: Late-Holocene bats of Mé Auré Cave, New Caledonia: Evidence of human consumption and a new species record from the recent past. The Holocene 22(1): 79-90. 2011
The Great Kai Island Giant Rat was described in 1923, it is known from three specimens that had been collected on the island of Kai Besar in the New Guinean part of Indonesia.
The species reached a total length of about 36 cm including the relatively short tail.
The Great Kai Island Giant Rat is sometimes treated as a subspecies of the White-tailed Giant Rat (Uromys caudimaculatus (Krefft)) or is considered synonymous to the Aru White-tailed Giant Rat (Uromys caudimaculatus ssp. aruensis (Laurie & Hill)).
The Chadwick Beach Cotton Deermouse was a subspecies of the Cotton Deermouse (Peromyscus gossypinus (Le Conte)) that was restricted to a small area along the coast of southwestern Sarasota County in Florida, USA.
The subspecies was described in 1939, apparently from specimens collected one year earlier, it was never found again and is now thought to be extinct.
Galapagos Rice Rat (Aegialomys galapagoensis ssp. galapagoensis)
“In 1835, when Darwin visited the Galapagos Islands, he found a native mouse inhabiting Chatham Island [Isla San Cristóbal] and supposed it to be the only indigenous mammal of the islands. This species was described as Mus galapagoensis by Waterhouse … who adds Darwin’s notation as follows: “This mouse or rat is abundant in Chatham Island. I could not find it on any other island of the group.” From this it is evident that Darwin made an effort to obtain further rodents, but his narrative seems to indicate that he did not spend any time on Narborough and Indefatigable islands, the principal ones from which specimens have been taken subsequently.” 
The Galapagos Rice Rat is one of several virtually unknown endemic rodent species that inhabit, or inhabited, the Galápagos Islands.
This species, which might include two subspecies, is known from at least two, maybe three, of the islands, with Isla San Cristóbal having been inhabited by the nominate, which was endemic to that island.
This form was apparently last collected in 1855 by Charles Darwin himself during the second voyage of HMS Beagle, it must have gone extinct only some decades later and all subsequent findings were of subfossil remains only.
 Wilfred H. Osgood: A new rodent from the Galapagos Islands. Field Museum of Natural History 17(2): 21-24. 1929
This is an up to date undescribed and thus unnamed species that is known exclusively from subfossil remains that had been found on Isla Isabele, Galápagos Islands. 
The Isabela sp. is apparently sometimes considered conspecific with the Galapagos Giant Rice Rat (Megaoryzomys curioi (Niethammer)) from the Isla Santa Cruz, another extinct rice rat species known only from subfossil remains.
 Gustavo Jiménez-Uzcátegui; Bryan Milstead; Cruz Márquez; Javier Zabala; Paola Buitrón; Alizon Llerena; Sandie Salazar; Birgit Fessl: Galapagos vertebrates: endangered status and conservation actions. Galapagos Report 2006-2007
The Guadalcanal Giant Rat, described in 1904, was endemic to the island of Guadalcanal, Solomon Islands, it is known from a single male specimen that had been cought sometimes between 1886 and 1888.
The species shared its habitat with another species of the same genus, the Emperor Rat (Uromys imperator (Thomas)), in contrast to that species, the Guadalcanal Giant Rat was not a giant at all, it reached a total length of only 35 cm, being half he size of its larger congener.
The Guadalcanal Giant Rat is now most likely extinct, the main reason for this are the same as in its congeneric ‘cousin’ – predation by introduced feral cats.
The Bluebuck was described in 1766; during the Late Pleistocene, the species was very widespread in South Africa, but the populations began to shrink following climatical changes at the beginning of the Holocene, in historical times, the species was finally restricted to a relict population that inhabited a rather small region at the southern tip of South Africa.
The species had a generally bluish gray fur, the upper lip and a patch in front of the eye were somewhat lighter, the forehead was brown, the belly was whitish.
The Bluebuck was hunted to extinction by European settlers, it is actually the first African mammal to become extinct in historical times, the last individuals were killed in 1799 or 1800.
Depiction from: ‘Francis Harper: Extinct and Vanishing Mammals of the Old World. New York, American Committee for international wildlife protection. Special Publication No. 12. 1945’
North African Hartebeest (Alcelaphus buselaphus ssp. buselaphus)
The North African Hartebeest [which is a monotypic species in my opinion] was once widely distributed all over northern Africa, including Egypt, Algeria, Jordan, Libya, and Morocco, and occurred probably also in Palestine and some parts of Saudi Arabia.
The animal reached a shoulder height of up to 1,5 m.
The North African Hartebeest was already well known to the ancient Egyptians and the Romans and can be found depicted Roman mosaics.
The species begun to disappear during the second half of the 19th century – due to direct hunting, especially during the French conquest of Algeria between 1830 and 1847, when thousands of the antelopes were killed by French soldiers just for amusement.
The last known individual was shot in the year 1925 in Morocco.
Yet, the species may have survived in the wild for a longer time than previously known – some bones, found 1998 in the Dakhla Oasis in Egypt inside of an abandoned hyena burrow could be dated to about 1960.
 A. J. Mills: Report presented to the Supreme Council of Antiquities, Egypt, on the 2000 Season of the Dakhleh Oasis Projekt. Dakhleh Oasis Project SCA Report: 1999/2000 Season
The Kenyan Oribi is one of the about nine subspecies of the Oribi (Ourebia ourebi Zimmerman), a small antelope species that is distributed over several parts of central-, eastern- and even southern Africa; it was apparently restcicted to the slopes of Mt. Kenya.
This form apparently died out in the wild sometimes during the 1960s and was subsequently kept in captivity for some time; its actual extinction date obviously is not known but is believed to be sometimes during the 1980s.
Oribi (Ourebia ourebi), nominate form?
Depiction from: ‘P. L. Sclater; O. Thomas: The Book of Antelopes. London: R. H. Porter 1894-1900’
The Twisted-toothed Mouse was described in 1929 based on subfossil remains, mainly teeth, which were recovered from cave deposits found both in the Dominican Republic as well as in Haiti, Hispaniola.
The species must have reached a size of 60 to 80 cm in length and might have weighed as much as about 20 kg.
There are contemporary reports from the 16th century which tell us of an animal called Quemi that was hunted and eaten by the native people of Hispaniola; these reports may refer to this species or maybe to the Hispaniolan Hutia (Plagiodontia aedium (F. Cuvier), a species that still exists.
The Laloumena Hippopotamus was described in 1990, it is the largest of the three known Malagasy Hippopotamus species, reaching almost the size of an African Hippopotamus (Hippopotamus amphibius L.).
This species is known from Pleistocene deposits that can be dated to an age of around 20000 years before present, but also from much younger remains that clearly are of Holocene age. 
It is quite possible that this species begun to disappear due to changes in the climatical conditions of Madagascar, leading to a drier climate, but it was very likely finally wiped out by humans.
 M. Faure; C. Guérin: Hippopotamus laloumena nov. sp., la troisième espèce d’hippopotame holocène de Madagascar. Comptes Rendus de l’Académie des Sciences. Ser. II 310: 1299-1305. 1990  Martine Faure; Claude Guérin; Dominique Genty; Dominique Gommery; Beby Ramanivosoa: Le plus ancien hippopotame fossile (Hippopotamus laloumena) de Madagascar (Belobaka, Province de Mahajanga). Comptes Rendus Palevol 9(4): 155-162. 2010
Large Montserrat Rice Rat (Cricetidae gen. & sp. A)
The Large Montserrat Rice Rat is an up to day undescribed species whose subfossil remains were found at an unnamed archaeological site on the island of Montserrat in the Lesser Antilles.
The species was sympatric with another closely related, yet smaller species which is likewise only known from subfossil remains. 
 Gregory K. Pregill; David W. Steadman; David R. Watters: Late Quaternary vertebrate faunas of the Lesser Antilles: historical components of Caribbean biogeography. Bulletin of Carnegie Museum of Natural History 30: 1-51. 1994
The Giant Island Deermouse was described in 1936 based on subfossil remains, the species was restricted to San Miguel – and Santa Rosa Islands, two of the Channel Islands offshore southwestern California, USA.
The species is believed to have disappeared sometimes between 2000 years ago and 1860, probably due to the accidental introduction of the North American Deermouse (Peromyscus maniculatus (Wagner)) by the Chumash, the native people of that region.
When Amerigo Vespucci, an Italian merchant, explorer, and navigator that lived from 1451 to 1512 and
from whose name the term “America” is derived, set foot on the island of Fernando de Noronha in 1503, he reported of: “marine and land birds without number” and also “very large rats.” 
This rat species had not been properly identified until subfossil remains were found abundantly in the 1970s.:
“The most frequently encountered fossils were those of a large rodent of the family Cricetidae, very different from the recent remains of the introduced rats (Rattus), which were also encountered on the dunes. This cricetid is almost certainly the rat mentioned by Vespucci. It is a new species, and possibly a new genus in the subfamily Sigmodontinae.” 
The species was finally described in 1999; it very likely died out very soon after the discovery of the island accompanied by the accidental introduction of rats. 
 S. L. Olson: Natural history of vertebrates on the Brazilian islands of the mid South Atlantic. National Geographic Society Research Reports 13: 481-492. 1981  Michael D. Carleton; Storrs L. Olson: Amerigo Vespucci and the rat of Fernando de Noronha: a new genus and species of Rodentia (Muridae, Sigmodontinae) from a volcanic island off Brazil’s continental shelf. American Museum Novitates 3256: 1-59. 1999
The Balearic Shrew, described in 1945, is known only from subfossil remains that were found on the island of Majorca and Menorca in the Balearic Islands, Spain; these remains can be dated to an age of about 2500 BC..
This species was described in 2011 and was originally known only from the type, a subadult individual that had been collected in 1979 in a fragment of moist forest on the western slopes of the Ecuadorian Andes.
A second specimen, discovered some years later in a museum collection was collected in 1959 at La Guayacana in western Colombia.
The Small Whiskered Bat apparently is/was restricted to the so-called Chocó ecoregion, lowland forest areas that now are mostly deforested, and is probably extinct. 
 Ricardo Moratelli; Don E. Wilson: A second record of Myotis diminutus (Chiroptera: Vespertilionidae): its bearing on the taxonomy of the species and discrimination from M. nigricans. Proceedings of the Biological Society of Washington 127(4): 533-543. 2015
Just like its next relative, the Oriente Cave Rat (Boromys offella Miller), this smaller spiny rat species is known only from subfossil bone remains, that had been found in several caves on the island of Cuba and on the Isla de la Juventud.
The reasons for its extinction are exactly the same as for its larger relative. 
 A. van der Geer; G. Lyras; J. de Vos; M. Dermitzakis: Evolution of Island Mammals: Adaptation and Extinction of Placental Mammals on Islands. John Wiley & Sons 2010
The Saudi Gazelle, described in 1935, was native to the Arabian Peninsula, where it inhabited sandy acacia plains.
The species disappeared du to excessive hunting and was declared extinct in the wild in 1980 when it very probably was already globally extinct, because all captive individuals were subsequently shown to represent hybrids or different species.
La Guarda Deermouse (Peromyscus guardia ssp. guardia)
The La Guarda Deermouse, also known as Angel Island Mouse, was described in 1912, the species is restricted to the Isla Ángel de la Guarda and several of the nearby smaller islets in the Gulf of California, Baja California, Mexico.
The nominate form inhabited the largest of the islands, Isla Ángel de la Guarda.
The La Guarda Deermouse with all its subspecies is now considered extinct, it fell victim to predation by introduced feral cats as well as competition by likewise introduced House Mice (Mus musculus L.), which now are found all over these islands.
This species was described in the year 1916 from subfossil bones that had been found on the island of Cuba and on the Isla de la Juventud.
The Oriente Cave Rat seems to have survived long enough to see the arrival of the first Europeans on the American double continent, because its bones were found in deposits that also contained bones of rats, which again reached Cuba for the first time together with the European discoverers.
These rats then again obviously played a big role in the extinction of this endemic rodent species. 
 A. van der Geer; G. Lyras; J. de Vos; M. Dermitzakis: Evolution of Island Mammals: Adaptation and Extinction of Placental Mammals on Islands. John Wiley & Sons 2010
The island of Tilos is located between the islands of Kos and Rhodes in the eastern Aegean Sea.
This island, like all of the islands in the Mediterranean Sea, once harbored its own, endemic fauna, including dwarfed animals like this endemic Elephant, whose fossil remains were already discovered in the 1970s.
The Tilos Dwarf Elephant was slightly larger than most other dwarf elephants known from other Mediterranean islands, it reached a shoulder height of about 1,2 to 1,6 m and a length of about 1,9 m.
The species died out around 3500 B.P., meaning it survived at least until the beginning of the Aegean Bronze Age. It was probably directly hunted to extinction. 
 G. E. Theodorou; N. Symeonides; E. Stathopoulou: Elephas tiliensis n. sp. from Tilos island (Dodecanese, Greece). Hellenic Journal of Geosciences 42: 19–32. 2007
The Lesser Pygmy Flying Squirrel was described in 1908 based on a single specimen that was collected in 1901 in Sabah, northern Borneo.
It is the smallest of the three species within its genus, reaching a full size of about 14 cm (including the tail).
The species was never found since its description and, probably being restricted to intact forest habitats, may now be extinct since large areas in the region have been converted into oil palm plantations.
The Nullabor Dwarf Bettong was described in 1997 based on subfossil skeletal remains that were found in caves on the Nullabor Plain, an large arid desert region in southern Australia.
The species apparently disappeared shortly after the arrival of European settlers in the region, who brought with them cats and foxes which preyed upon the native mammals and still do so up to this day.
The native people of the Pilbara region allegedly have two names for a very small kangaroo species, weelba respectively wirlpa, which may have originally been used for this species. 
 Chris Johnson: Australia’s Mammal Extinctions: a 50000 year history. Cambridge University Press 2006
Indeterminate Hutia (Capromyidae gen. & sp. ‘Hispaniola’)
The subfossil remains of this form, whose closest relative seems to be the Imposter Hutia (Hexolobodon phenax (Miller)), known only in a subfossil state as well, were found in 1989 (?) on the island of Hispaniola.
The bones have an age of about 3600 to 4700 years. 
 Samuel T. Turvey: Holocene Extinctions. Oxford University Press, USA 2009
This species was described in 1839 on the basis of fossil remains that were dated to an Late Pleistocene age.
The species survived into the early Holocene era.
 Andrea Corona; Daniel Perea; H. Gregory McDonald: Catonyx cuvieri (Xenarthra, Mylodontidae, Scelidotheriinae) from the late Pleistocene of Uruguay, with comments regarding the systematics of the subfamily. Journal of Vertebrate Paleontology 33(5): 1214-1225. 2013
Crooked Island Hutia (Geocapromys ingrahami ssp. irrectus)
The Bahaman Hutia (Geocapromys ingrahami (J. A. Allen)) is a species of middle-sized rodent that is endemic to the Bahamas, or at least to the islands that are part of the so-called Great Bahama Bank.
The species was believed to be extinct, but a very small population was rediscovered in 1966 on a likewise very small island named East Plana Cay, this was assigned to the nominate form and in 1973, some of these animals were released on two additional islands, Little Wax Cay and Warderick Wells.
Based on DNA studies it is now known that the animals on East Plana Cay in fact did not constitute the last surviving wild population but one that was introduced in pre-Columbian times by the people of the Lucayan culture, the first human inhabitants of the Bahamas. 
Subfossil remains of this species were found on many of the larger islands, including Abaco as well as Crooked Island, Eleuthera, Exuma and Long Island; these forms differed from the alleged nominate form by their cranial characters and thus were described as two distinct subspecies: Geocapromys ingrahami ssp. abacoensis Lawrence (from Abaco Island) and Geocapromys ingrahami ssp. irrectus (from the remaining islands); the form from Abaco Island, however, is now known to have also just been introduced in prehistoric times, thus has never been a distinct subspecies. 
The animals that inhabited Crooked Island, Eleuthera, Exuma and Long Island on the other hand, appear to be non-monophyletic, thus likewise do not represent a distinct subspecies but either more than one or none at all.
The above-mentioned differences in the cranial characters may in fact just be the consequence of resource availability – animals on larger islands may have become larger because they had access to larger resources …. 
The Bahamian Hutia, once believed to be extinct, then rediscovered in 1966, may never had have any subspecies after all. This question is not yet answered – I will mention this species here only fro the sake of completness and because it constitutes a very interesting case.
 B. N. Lawrence: Geocapromys from the Bahamas. Occasional papers of the Boston Society of Natural History 8: 189-196. 1934  Jessica A. Oswald; Julie M. Allen; Michelle J. LeFebvre; Brian J. Stucky; Ryan A. Folk; Nancy A. Albury; Gary S. Morgan; Robert P. Guralnick; David W. Steadman: Ancient DNA and high-resolution chronometry reveal a long-term human role in the historical diversity and biogeography of the Bahamian hutia. Scientific Reports 10: 1373. 2020
The Haitian Edible Rat was described in 1929 based on subfossil remains.
In pre-European times, the native spiny rats of the Caribbean were very common and were an important part of the diet of the indigenous people.
Most of the known species – including this one – did not become extinct until a short time after the arrival of the Europeans. 
 Gerrit S. Miller: A second collection of mammals from caves near St. Michel, Haiti. Smithsonian Miscellaneous Collections 81(9): 1-30. 1929
 Samuel T. Turvey: Holocene Extinctions. Oxford University Press, USA 2009
 Alexandra van der Geer; George Lyras; John de Vos; Michael Dermitzakis: Evolution of Island Mammals: Adaptation and Extinction of Placental Mammals on Islands. John Wiley & Sons 2010
Photo from: ‘Gerrit S. Miller: A second collection of mammals from caves near St. Michel, Haiti. Smithsonian Miscellaneous Collections 81(9): 1-30. 1929’
Queen of Sheba’s Gazelle aka. Yemen Gazelle was for some time believed to be a subspecies of the Arabian Gazelle (Gazella arabica (Lichtenstein)) (see photo below) but is now considered a distinct species.
The species inhabited the mountains near city of Ta’izz in southwestern Yemen where it was last seen in 1951 when it apparently still was quite common.
In 1985, a photograph of gazelles taken in a private collection, the Al Wabra Wildlife Farm in Qatar, might show this species, however, this has apparently never been confirmed, nor seems the subsequent fate of these animals to be known.
Queen of Sheba’s Gazelle is now considered extinct with the cause of its extinction being uncertain, however, it most likely was hunted to extinction.
Montserrate Island Pocket Mouse (Chaetodipus rudinoris ssp. fornicatus)
The Montserrat Island Pocket Mouse is one of six subspecies of Baja Pocket Mouse (Chaetodipus rudinoris (Elliot)), it was restricted to the Isla Montserrate, a small island in the Gulf of California, Mexico.
The subspecies disappeared due to predation by introduced feral cats; it was last recorded in 1975 and is now considered extinct.
The Pallid Beach Deermouse was a subspecies of the Oldfield Deermouse (Peromyscus polionotus (Wagner)) that inhabited a small area of coastal sand dunes along the Atlantic coast of Florida, USA.
This form was quite abundant when it was described in 1939, however the populationds disappeared subsequently and no individual was found ever since, thus this subspecies is now redarded as being extinct.
The Short-tailed Hopping Mouse was described in 1936 based on two female specimens that were collected already in 1896; it inhabited open, stony plains with a vegetation dominated by grasses and low shrubs in the vicinity of the town of Alice Springs in central Australia.
The species apparently disappeared due to predation by introduces cats and foxes, combined with habitat destruction.
The Granito Deermouse, described in 1967, was a subspecies of the Guarda Deermouse (Peromyscus guardia Townsend) and was restricted to the small Isla Granito in the Gulf of California, Baja California, Mexico.
The whole species, including its three named and one unnamed subspecies, is now extinct.