Short answer would be no, but many people forget that Tolkien made allusions to many unnamed creatures outside of known classifications. Morgoth bred ,,monsters of diverse kinds" and some unspecified monsters appeared from side effect of his malice flowing into the world during the Spring of Arda, as for the classification of sapient races: we have elves, dwarves, men (Druedain may be special subspecies of mankind as they appear to be very much like short, neandertal-like people, possibly connected with the myths about woodwoses), halflings, orcs, trolls, ents, mysterious Stone Giants, then come different ,,non-humanoid" beings of high intelligence that could be considered semi-sentient: wargs, giant spiders, dragons, Great Eagles, werewolves, possibly bat-like vampires, even more vague nameless things gnawing the world, Watcher in the Water, there are also typical non-intelligent beasts like flying fell beasts also called hell-hawks, besides that there are various undead beings and evil spirits and ,,other things hostile to men and elves" as Middle Earth is ,,full of strange creatures beyond count", there is also very ambiguous and possibly just legendary case of ogres mentioned in The Hobbit along with references to creatures living underground in goblin caves of which even they didn't know about:
,,He could not swim; and he thought, too, of nasty slimy things, with big bulging blind eyes, wriggling in the water. There are strange things living in the pools and lakes in the hearts of mountains: fish whose fathers swam in, goodness only knows how many years ago, and never swam out again, while their eyes grew bigger and bigger and bigger from trying to see in the blackness; also there are other things more slimy than fish. Even in the tunnels and caves the goblins have made for themselves there are other things living unbeknown to them that have sneaked in from outside to lie up in the dark. some of these caves, too, go back in their beginnings to ages before the goblins, who only widened them and joined them up with passages, and the original owners are still there in odd corners, slinking and nosing about." There is also this thing:
,,...mounted upon a black horse, if horse it was; for it was huge and hideous, and its face was a frightful mask, more like a skull than a living head, and in the sockets of its eyes and in its nostrils there burned a flame" and again reference to dark creatures of unknown origin:
,,And there were murmured hints of creatures more terrible than all these (orcs and trolls), but they had no name.").
So in a way we know there is no explicit mention of unicorns (though Mearas are pretty magical and could form a substitute, in Aman there were apparently all kinds of life forms that live and ever lived in Arda together with quite unique species that were never seen outside of the Blessed Realm, also some otherwise ordinary animals are enhanced and raised to higher level given special powers by the Valar), phoenixes, basilisks (unless there were some reptiles, ,,creatures of the older world" like fell beast steeds of Nazgul that could resemble one) and all that typical stuff from myths (though in early drafts Book of Lost Tales there appear mermaids or local version of those, known as Oarni spirits of the sea that were probably lesser than Maiar serving Ulmo or something else entirely fairy creatures that later were simply forgotten by Tolkien in more refined version of stories of that Goldberry the River Daughter might be a leftover) in no way we can say that Middle Earth or rather world of Arda lacks mystery or is as well known as most would think.