also dish-pan, "pan in which dishes are washed," 1858, from dish (n.) + pan (n.). Dishpan hands "inflamed or sore hands caused by housework" is attested by 1935, an advertiser's phrase.
In 1922, Lever Brothers began to advertise Lux in this country to "rid your hands of that dishpan look." Without any break since then the company has alluded to "dishpan hands" which come from using soap that is too strong in alkaline content. [Printers' Ink, vol. 173, 1935]