  If $G$ is a finite $\ell$-group acting on an affine space $\mathbb{A}^n$ over
a finite field $K$ of cardinality prime to $\ell$, Serre has shown that there
exists a rational fixed point. We generalize this to the case where $K$ is a
henselian discretely valued field of characteristic zero with algebraically
closed residue field and with residue characteristic different from $\ell$. We
also treat the case where the residue field is finite of cardinality $q$ such
that $\ell$ divides $q-1$. To this aim, we study group actions on weak N\'eron
models.
