Well, maybe I should let other, better versed in Tolkien lore 'downers reply to this point, but if you will forgive a lack of textual evidence here's what happens (according to my understanding):
The ring is not quite an unlimited source of power. It is said that the ring gives people (or hobbits, or maiar, or deers) power according to their stature. So a hobbit will probably have little use of the ring for anything else than dissapearing unless this hobbit also spent most of his lifetime learning how to wield the powers of the ring. And it's still likely that even then he would not be able to unleash it's full power.
On the other hand, a powerful elf would be able to gain more power from it, but probably not as much power as a Maiar such as Gandalf or Sauron himself. With your example of a deer, the deer might have some super-natural powers, but by no means it'd be an allmighty little cute deer (yet I see how amusing that could be
)
Also, the deer will probably lack the intelligence to keep the ring, as it tends to slip off when it wants to change masters, and it's likely that the Deer would not know how to avoid it from slipping onto a river or something.
But the last part is mere speculation.