We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued field
K
\mathbb {K}
. To capture the features of classical amenability that induce the vanishing of bounded cohomology with real coefficients, we start by introducing the notion of normed
K
\mathbb {K}
-amenability, of which we prove an algebraic characterization. It implies that normed
K
\mathbb {K}
-amenable groups are locally elliptic, and it relates an invariant, the norm of a
K
\mathbb {K}
-amenable group, to the order of its discrete finite
p
p
-subquotients, where
p
p
is the characteristic of the residue field of
K
\mathbb {K}
. Moreover, we prove a characterization of discrete normed
K
\mathbb {K}
-amenable groups in terms of vanishing of bounded cohomology with coefficients in
K
\mathbb {K}
.
The algebraic characterization shows that normed
K
\mathbb {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
\mathbb {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
\mathbb {K}
has positive characteristic or its residue field has characteristic
0
0
, then the comparison map is injective in all degrees. If
K
\mathbb {K}
is a finite extension of
Q
p
\mathbb {Q}_p
, we classify unbounded and non-trivial 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; this applies in particular to finitely presented groups in degree
2
2
.
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, the main missing piece for computing real bounded cohomology.