In this article the following are proved: 1. Let G be an infinite p-group of cardinality either No or greater than 2(N0). If G is center-by-finite and non-Cernikov, then it is non-co-Hopfian; that is, G is isomorphic to a proper subgroup of itself. 2. Let G be a nilpotent p-group of class 2 with G/G' a non-Cernikov group of cardinality No or greater than 2(N0). If G' is of order p, then G is non-co-Hopfian.