  We consider linear groups and Lie groups over a non-Archimedean local field
$\mathbb F$ for which the power map $x\mapsto x^k$ has a dense image or it is
surjective. We prove that the group of $\mathbb F$-points of such algebraic
groups is a compact extension of unipotent groups with the order of the compact
group being relatively prime to $k$. This in particular shows that the power
map is surjective for all $k$ is possible only when the group is unipotent or
trivial depending on whether the characteristic of $\mathbb F$ is zero or
positive. Similar results are proved for Lie groups via the adjoint
representation. To a large extent, these results are extended to linear groups
over local fields and global fields.
