  Motivated by Kloeckner's result on the isometry group of the quadratic
Wasserstein space $\mathcal{W}_2\left(\mathbb{R}^n\right)$, we describe the
isometry group $\mathrm{Isom}\left(\mathcal{W}_p (E)\right)$ for all parameters
$0 < p < \infty$ and for all separable real Hilbert spaces $E.$ In fact, the
$0<p<1$ case is a consequence of our more general result: we prove that
$\mathcal{W}_1(X)$ is isometrically rigid if $X$ is a complete separable metric
space that satisfies the strict triangle inequality. Furthermore, we show that
this latter rigidity result does not generalise to parameters $p>1$, by solving
Kloeckner's problem affirmatively on the existence of mass-splitting
isometries.
