  We study continuous bounded cohomology of totally disconnected locally
compact groups with coefficients in a non-Archimedean valued field $K$. To
capture the features of classical amenability that induce the vanishing of real
bounded cohomology, we introduce the notion of normed $K$-amenability, of which
we prove an algebraic characterization. It implies that normed $K$-amenable
groups are locally elliptic, and it relates an invariant, the norm of a
$K$-amenable group, to the order of its discrete finite $p$-subquotients, where
$p$ is the characteristic of the residue field of $K$. Moreover, we prove a
bounded-cohomological characterization for discrete groups.
  The algebraic characterization shows that normed $K$-amenability is a very
restrictive condition, so the bounded cohomological one suggests that there
should be plenty of groups with rich bounded cohomology with trivial $K$
coefficients. We explore this intuition by studying the injectivity and
surjectivity of the comparison map, for which surprisingly general statements
are available. Among these, we show that if either $K$ has positive
characteristic or its residue field has characteristic 0, then the comparison
map is injective in all degrees. If $K$ is a finite extension of
$\mathbb{Q}_p$, we classify quasimorphisms of a group and relate them to its
subgroup structure. For discrete groups, we show that suitable finiteness
conditions imply that the comparison map is an isomorphism.
  A motivation as to why the comparison map is often an isomorphism, in stark
contrast with the real case, is given by moving to topological spaces. We show
that over a non-Archimedean field, bounded cohomology is a cohomology theory in
the sense of Eilenberg--Steenrod, except for a weaker version of the additivity
axiom which is however equivalent for finite disjoint unions. In particular
there exists a Mayer--Vietoris sequence.
