Reprinted from the "Pitt Panther," a student publication of the University of Pittsburgh, this just may be the earliest reference currently known:

Trenton [N.J.] Sunday Times-Advertiser (Nov. 24, 1929): College Humor...Mexican Pete: "Pardon me, Deadeye Dick, may I borrow your lighter?"

Caveat: "Deadeye Dick" and "Mexican Pete" were fairly common Wild West nicknames. Still, it's startling to see them used together at a time when "Eskimo Nell" was likely in circulation.

The earliest clear reference to "Eskimo Nell" I have found is this, from Smith’s Weekly (Sydney, Aus.), Oct. 29, 1932: “A Central Australian Wild West has grown up overnight…. Dynamite Annie and Eskimo Nell are there in all their ferocious finery.”

The psychologist H. J. Eysenck, in his autobiography, recalled
hearing the poem as a student at the University of London the same year.