  We show that if $G$ is a non-archimedean, Roelcke precompact, Polish group,
then $G$ has Kazhdan's property (T). Moreover, if $G$ has a smallest open
subgroup of finite index, then $G$ has a finite Kazhdan set. Examples of such
$G$ include automorphism groups of countable $\omega$-categorical structures,
that is, the closed, oligomorphic permutation groups on a countable set. The
proof uses work of the second author on the unitary representations of such
groups, together with a separation result for infinite permutation groups. The
latter allows the construction of a non-abelian free subgroup of $G$ acting
freely in all infinite transitive permutation representations of $G$.
