E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
To finish a thing there and then without stopping; to make a clean sweep of a job in hand. Judge Haliburtons Sam Slick popularised the word. (German, schlicht, sleek, polished, hence clean; Icelandic, slke, sleek.) We say, To do a thing clean off as well as slick off.