  A K3 surface $X$ over a $p$-adic field $K$ is said to have good reduction if
it admits a proper smooth model over the ring of integers of $K$. Assuming
this, we say that a subgroup $G$ of $\mathrm{Aut}(X)$ is extendable if $X$
admits a proper smooth model equipped with $G$-action (compatible with the
action on $X$). We show that $G$ is extendable if it is of finite order prime
to $p$ and acts symplectically (that is, preserves the global $2$-form on $X$).
The proof relies on birational geometry of models of K3 surfaces, and
equivariant simultaneous resolutions of certain singularities. We also give
some examples of non-extendable actions.
