mid-15c., creke "narrow inlet in a coastline," altered from kryk (early 13c.; in place names from 12c.), probably from Old Norse kriki "corner, nook," perhaps influenced by Anglo-French crique, itself from a Scandinavian source via Norman. Perhaps ultimately related to crook and with an original notion of "full of bends and turns" (compare dialectal Swedish krik "corner, bend; creek, cove").
Extended to "inlet or short arm of a river" by 1570s, which probably led to use for "small stream, brook" in American English (1620s). In U.S. commonly pronounced and formerly sometimes spelled crick. Also used there and in Canada, Australia, New Zealand for "branch of a main river," possibly from explorers moving up main rivers and seeing and noting mouths of tributaries without knowing they often were extensive rivers of their own.
Slang phrase up the creek "in trouble" (often especially "pregnant") is attested by 1941, perhaps originally armed forces slang for "lost while on patrol."