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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 (cf. blink).
Very fast: Felt he just had to get a lawyer lickity-split
[1859+; fr lick, ''speed, a spurt of speed,'' found by 1809; earlier forms lickety-cut, lickety-click, lickety liner, and lickety switch are found in the 1830s and 1840s]