"a stick, a club," Old English *batt "cudgel," perhaps from Celtic (cf. Irish and Gaelic bat, bata "staff, cudgel"), influenced by Old French batte, from Late Latin battre "beat;" all from PIE root *bhat- "to strike." Also "a lump, piece" (mid-14c.), as in brickbat. As a kind of paddle used to play cricket, it is attested from 1706.
flying mammal (order Chiroptera), 1570s, a dialectal alteration of Middle English bakke (early 14c.), which is probably related to Old Swedish natbakka, Old Danish nathbakkæ "night bat," and Old Norse leðrblaka "leather flapper," so original sense is likely "flapper." The shift from -k- to -t- may have come through confusion of bakke with Latin blatta "moth, nocturnal insect."
Old English word for the animal was hreremus, from hreran "to shake." As a contemptuous term for an old woman, perhaps a suggestion of witchcraft (cf. fly-by-night), or from bat as "prostitute who plies her trade by night" [Farmer, who calls it "old slang" and finds French equivalent "night swallow" (hirondelle de nuit) "more poetic"].
"to move the eyelids," 1847, American English, from earlier sense of "flutter as a hawk" (1610s), a variant of bate (v.2) on the notion of fluttering wings. Related: Batted; batting.