A (left) Engel sink of an element g of a group G is a subset containing all sufficiently long commutators \([...[[x,g],g],\dots ,g]\) , where x ranges over G. We prove that if p is a prime and G a finite group in which, for some positive integer m, every p-element has an Engel sink of cardinality at most m, then G has a normal subgroup N, such that G/N is a \(p'\) -group and the index \([N:O_p(G)]\) is bounded in terms of m only. Furthermore, if G is a profinite group in which every p-element possesses a finite Engel sink, then G has a normal subgroup N such that N is virtually pro-p, while G/N is a pro- \(p'\) group.
