  Let $X$ be a locally compact Hadamard space and $G$ be a totally disconnected
group acting continuously, properly and cocompactly on $X$. We show that a
closed subgroup of $G$ is amenable if and only if it is (topologically locally
finite)-by-(virtually abelian). We are led to consider a set $\bdfine X$ which
is a refinement of the visual boundary $\bd X$. For each $x \in \bdfine X$, the
stabilizer $G_x$ is amenable.
