 Elements that are powers of infinitely many degrees in a local field

For a non-Archimedean local field $K$, an element of $K^{\times }$ is an $m$th power for infinitely many positive integers $m$ exactly when it belongs to ${\mathcal{O}}_K^{\times }$. Valuations prove necessity; for sufficiency, choose infinitely many $m$ coprime to the residue characteristic and to $|k^{\times }|$, for which exponentiation by $m$ is an automorphism of the unit group.