Old English deor "wild animal, beast, any wild quadruped," in early Middle English also used of ants and fish, from Proto-Germanic *deuzam, the general Germanic word for "animal" (as opposed to man), but often restricted to "wild animal" (source also of Old Frisian diar, Dutch dier, Old Norse dyr, Old High German tior, German Tier "animal," Gothic dius "wild animal," also see reindeer).
This is perhaps from PIE *dheusom "creature that breathes," from root *dheu- (1) "cloud, breath" (source also of Lithuanian dusti "gasp," dvėsti "gasp, perish;" Old Church Slavonic dychati "breathe"). For possible prehistoric sense development, compare Latin animal from anima "breath").
The sense specialization to a specific animal began in Old English (the usual Old English word for what we now call a deer was heorot; see hart), was common by 15c., and is now complete. It happened probably via hunting, deer being the favorite animal of the chase (compare Sanskrit mrga- "wild animal," used especially for "deer").
Deer-lick "salty spot where deer come to lick," is attested by 1778, in an American context. The deer-mouse (1840) is so called for its agility.