Humans have in fact been arguing about which animals lie down to sleep for more than two millennia, particularly disagreeing over which are are physically incapable. In 350 BC an apparently irritated Aristotle firmly debunked the myths that elephants were incapable of lying down: “what is asserted of the elephant by some is not true (ie that he cannot sit or bend his knees) for he can do both.” By the sounds of it, the argument had been raging for some time.
In fact, almost any creature does seem to be able to lie down in some fashion, even giraffes which do so rarely and with some difficulty. Most scientists agree that lying in a relaxed position is essential to getting the deep, replenishing REM sleep that mammals definitely require. Remaining standing requires a level of muscular exertion and consciousness that simply doesn’t lend itself to deep sleep, though there are stories of exhausted humans involved in wars who could, in a fashion, sleep while standing or marching.
There are animals which choose not to lie down, however, especially birds. Flamingos often live on caustic salt flats, sometimes amidst water which is boiling, and where a short recline would mean certain death. They remain standing when sleeping, rather sensibly. Many other birds join them in spending most of their lives on their feet or at wing, only lying down to die. They appear to survive perfectly well without sleeping prostrate.
At least one common mammal also doesn’t lie down in typical circumstances – bats, which hang from trees and caves. However, bats’ claws automatically lock in place through body weight, which allows the rest of their bodies to go into relaxation. This may not be laying down but it physically performs almost exactly the same function of relieving all exertion and stress.
If you want to find a mammal that doesn’t lie down, you should look at the oceans, where whales and dolphins cannot allow themselves to rest on the ocean floor, given that they need to rise to the surface to breathe. Instead, they have evolved a kind of sleep called ‘unihemispheric’ or one-sided sleep, where they shut down one hemisphere of their brains and its corresponding eye, while the other half of the brain and eye stays alert. During these states they may well float or bob around, but never exactly lie down.
Once you are in the oceans there are of course an extraordinary array of new creatures from molluscs to fish who not only don’t lie down in the sense we mean on the land, but don’t sleep in the same way either.
In many cases, scientists still don’t understand exactly how these creatures do rest and whether they need it in the same way that mammals do. That, however, is a whole other article, and for another day.