  We characterize sandwiched singularities in terms of their link in two
different settings. We first prove that such singularities are precisely the
normal surface singularities having self-similar non-archimedean links. We
describe this self-similarity both in terms of Berkovich analytic geometry and
of the combinatorics of weighted dual graphs. We then show that a complex
surface singularity is sandwiched if and only if its complex link can be
embedded in a Kato surface in such a way that its complement remains connected.
