Green’s Dictionary of Slang

shit creek n.

[fig. use of shit n. (1a) + SE creek]

[1950s+] an unpleasant, problematic situation; usu. in phr. below.

In phrases

up shit(’s) creek (also down shit creek without a paddle, in it without a paddle, up salt creek (without a paddle), up shit creek and no paddle, up shit creek in leaky gumboots, up shit creek without a paddle (in a barbed-wire canoe, …in a leaky canoe), up shit creek without a paddle to piss on, up shit crick)

[mid-19C+] (orig. US) in serious trouble.