  We prove that a connected locally compact median space of finite rank which
admits a transitive action is isometric to $\mathbb{R}^n$ endowed with the
$\ell^1$-metric. In the other side, replacing the transitivity assumption on
the group of isometries by a certain regularity of the action on the
compactification of the space, we show that all orbits are discrete. In our way
to prove these results, we give a characterization of the compactness in
complete median spaces of finite rank by the combinatorics of their halfspaces.
