In this article the following are proved: 1. Let $G$ be an infinite $p$-group of cardinality either ${\bf {\mathbb N}_{0}}$ or greater than $2^{\bf {\mathbb N}_{0}}$. If $G$ is center-by-finite and non-$\skew5\check{C}$ernikov, 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-$\skew5\check{C}$ernikov group of cardinality ${\bf {\mathbb N}_{0}}$ or greater than $2^{{\bf {\mathbb N}_{0}}}$. If $G'$ is of order $p$, then $G$ is non-co-Hopfian.