Douglas Harper's Etymology Dictionary
1852, American English (earlier lickety-cut, lickety-click, and simply licketie, 1817), from lick (n.1) in dialectal sense "very fast sprint in a race" (1809) on the notion of a "lick" as a fast thing (compare blink).
Wiktionary
adv. quickly; as fast as possible; in short order.
Usage examples of "lickety-split".
I got what felt like a ear in my mouth and commenced chawing on it, and presently, what with this and other mayhem I committed on him, he give a most inhuman squall and bust away and went lickety-split for the outside world.
His words, though scarcely original, a medley of bleak cliches, had tapped into the core of my weakness, and all the fear I had nourished over long years in Africa, all the peculiarly African fear, the apprehension of spirits, of lion ghosts and magical presidents and the long-legged, licorice-skinned, lickety-split demons of the talking drum dancing with arms akimbo on the margins of one's campfire light, ready to pounce with their red-hot spears and golden teeth.