The aim of this paper is to prove that the $p$-Wasserstein space
$\mathcal{W}_p(X)$ is isometrically rigid for all $p\geq 1$ whenever $X$ is a
countable graph metric space. As a consequence, we obtain that for every
countable group $H$ and any $p\geq 1$ there exists a $p$-Wasserstein space
whose isometry group is isomorphic to $H$.Comment: 14 pages with 3 figure