1570s, "to pass in running," originally in Lyly, perhaps from out- + Middle English strip "move quickly, make a stroke" (in reference to a weapon). c. 1400, a word of uncertain origin, perhaps from stripe (n.). Or outstrip might be a corruption of outstrike (15c.), from strike (v.) in the old sense of "go, proceed, advance." The figurative sense of "to excel or surpass in anything" is from 1590s. Related: Outstripped; outstripping. The punning references to strip (v.) date from late 19c.
The abridged petticoats of the ladies proceeded, no doubt, to an intolerable pitch; and they tried, as Byron said, to outstrip one another. [W. Carew Hazlitt, "Four Generations of a Literary Family," 1897, referring to Henry James Byron, the dramatist and the author's friend, not Lord Byron, the poet]