Beavers were known in Anglo-Saxon times: the word is beofor. Then, in the 15th century, a French word, baviere, appears in English, referring to the lower part of the face-guard of a helmet, and in due course this came to be spelled and pronounced in the same way as the animal.
There is no instance of the animal in Shakespeare, but there are seven references to the piece of armour. There is never any real ambiguity, as the context is always one of war, not river-banks.
"He wore his beaver up," says Horatio of the Ghost (Hamlet, I.ii.230). "I saw young Harry with his beaver on," says Vernon of Prince Hal (Henry IV Part 1, IV.i.104). But it can nonetheless take (especially young) readers by surprise, and to avoid a surreal mental image, it is as well to draw attention to the older meaning.
When King Richard exclaims, "What, is my beaver easier than it was?"
(Richard III, V.iii.50), he is not enquiring after a rodent's state of health.