  Among connected linear algebraic groups, quasi-reductive groups generalize
pseudo-reductive groups, which in turn form a useful relaxation of the notion
of reductivity. We study quasi-reductive groups over non-archimedean local
fields, focusing on aspects involving their locally compact topology.
  For such groups we construct valuated root data (in the sense of
Bruhat--Tits) and we make them act nicely on affine buildings. We prove that
they admit Iwasawa and Cartan decompositions, and we construct small compact
open subgroups with an Iwahori decomposition.
  We also initiate the smooth representation theory of quasi-reductive groups.
Among others, we show that their irreducible smooth representations are
uniformly admissible, and that all these groups are of type I.
  Finally we discuss how much of these results remains valid if we omit the
connectedness assumption on our linear algebraic groups.
