The purpose of this paper is to link anisotropy properties of an algebraic
group together with compactness issues in the topological group of its rational
points. We nd equivalent conditions on a smooth ane algebraic group scheme over
a non-Archimedean local eld for the associated rational points to admit maximal
compact subgroups. We use the structure theory of pseudo-reductive groups
provided, whatever the characteristic, by Conrad, Gabber and Prasad. We also
investigate thoroughly maximal prop subgroups in the semisimple case, using
Bruhat-Tits theory
