'Still' is PIE *stel- (to stand firm) — the temporal 'continuing' sense grew from remaining unmoved.
Continuing up to the present time or the time mentioned; nevertheless; even so. Also an adjective meaning 'not moving' and a noun for a distillation apparatus.
From Old English 'stille' (motionless, calm, silent, fixed), from Proto-Germanic *stiljaz (standing firm, fixed, unmoving), from PIE *stel- (to put in place, to stand, to be firm). The PIE root carries the sense of deliberate placement or fixedness. The temporal adverb sense — 'he is still here' meaning he continues to be here — emerged in Middle English from the logic that what is motionless has not
The distillation apparatus called a 'still' comes from a completely different etymological path — it is a shortening of 'distill,' from Latin 'dēstillāre' (to drip down). Its resemblance to 'still' meaning 'motionless' is pure coincidence.