"to beat, surpass, overcome" 1530s, perhaps from figurative use of lick (v.1) in the Coverdale bible that year in a sense of "defeat, annihilate" (an enemy's forces) in Numbers xxii.4:
Now shal this heape licke up all that is about vs, euen as an oxe licketh vp the grasse in the field.
But to lick (of) the whip "taste punishment" is attested from mid-15c.
Old English liccian "to pass the tongue over the surface, lap, lick up," from Proto-Germanic *likkon (source also of Old Saxon likkon, Dutch likken, Old High German lecchon, German lecken, Gothic bi-laigon), from PIE root *leigh- "to lick."
French lécher, Italian leccare are said to be Germanic loan words. The figurative lick (one's) lips in eager anticipation is from c. 1500. Lick-ladle (1849) was an old phrase for a (human) parasite. To lick (someone or something) into shape (1610s) is in reference to the supposed ways of bears:
Beres ben brought forthe al fowle and transformyd and after that by lyckyng of the fader and the moder they ben brought in to theyr kyndely shap. ["The Pylgremage of the Sowle," 1413]
"an act of licking," c. 1600, from lick (v.1). The earlier noun was licking (late 14c.; Old English had liccung). The meaning "small portion" is 1814, originally Scottish; hence the U.S. colloquial sense. Sense of "place where an animal goes to lick salt" is from 1747. The jazz music sense of "short figure or solo" is by 1922, perhaps from an earlier colloquial sense "a spurt or brisk run in racing" (1809). Meaning "a smart blow" (1670s) is from lick (v.2).