Old English drincan "to drink," also "to swallow up, engulf" (class III strong verb; past tense dranc, past participle druncen), from Proto-Germanic *drenkan (source also of Old Saxon drinkan, Old Frisian drinka, Dutch drinken, Old High German trinkan, German trinken, Old Norse drekka, Gothic drigkan "to drink"), which is of uncertain origin or connections, perhaps from a root meaning "to draw." Most Indo-European words for this trace to PIE *po(i)- (source of Greek pino, Latin biber, Irish ibim, Old Church Slavonic piti, Russian pit'; see imbibe). The noun meaning "beverage, alcoholic beverage" was in late Old English.
The noun, AS. drinc, would normally have given southern drinch (cf. drench), but has been influenced by the verb. [Weekley]To drink like a fish is first recorded 1747.