  The aim of this short paper is to offer a complete characterization of all
(not necessarily surjective) isometric embeddings of the Wasserstein space
$\mathcal{W}_p(\mathcal{X})$, where $\mathcal{X}$ is a countable discrete
metric space and $0<p<\infty$ is any parameter value. Roughly speaking, we will
prove that any isometric embedding can be described by a special kind of
$\mathcal{X}\times(0,1]$-indexed family of nonnegative finite measures. Our
result implies that a typical non-surjective isometric embedding of
$\mathcal{W}_p(\mathcal{X})$ splits mass and does not preserve the shape of
measures. In order to stress that the lack of surjectivity is what makes things
challenging, we will prove alternatively that $\mathcal{W}_p(\mathcal{X})$ is
isometrically rigid for all $0<p<\infty$.
