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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 find equivalent conditions on a smooth affine algebraic group scheme over a non-Archimedean local field 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 pro-p subgroups in the semisimple case, using Bruhat–Tits theory.

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