  We study the geometry of the space of measures of a compact ultrametric space
X, endowed with the L^p Wasserstein distance from optimal transportation. We
show that the power p of this distance makes this Wasserstein space affinely
isometric to a convex subset of l^1. As a consequence, it is connected by
1/p-H\"older arcs, but any a-H\"older arc with a>1/p must be constant. This
result is obtained via a reformulation of the distance between two measures
which is very specific to the case when X is ultrametric; howeverthanks to the
Mendel-Naor Ultrametric Skeleton it has consequences even when X is a general
compact metric space. More precisely, we use it to estimate the size of
Wasserstein spaces, measured by an analogue of Hausdorff dimension that is
adapted to (some) infinite-dimensional spaces. The result we get generalizes
greatly our previous estimate that needed a strong rectifiability assumption.
The proof of this estimate involves a structural theorem of independent
interest: every ultrametric space contains large co-Lipschitz images of
\emph{regular} ultrametric spaces, i.e. spaces of the form {1,...,k}^N with a
natural ultrametric. We are also lead to an example of independent interest: a
space of positive lower Minkowski dimension, all of whose proper closed subsets
have vanishing lower Minkowski dimension.
