Can Cayley's Theorem be applied to groups with infinite order?

The group $\mathbb Z$ is isomorphic to the group of all permutations of $\mathbb Z$ of the form $m\mapsto m+n$ ($n\in\mathbb Z$).

More generally, if $G$ is any group, then $G$ is isomorphic to the group of all permutations of $G$ of the form $g\mapsto gh$ ($h\in G$).

Your professor is incorrect. Any group $G$ acts on itself by left multiplication, and this action gives an embedding of $G$ in $\operatorname{Sym}(G)$.

Thus every group is isomorphic to a subgroup of a group of permutations. And that is precisely what Cayley's theorem states.