  We consider a broad class of semilinear SPDEs with multiplicative noise
driven by a finite-dimensional Wiener process. We show that, provided that an
infinite-dimensional analogue of H\"ormander's bracket condition holds, the
Malliavin matrix of the solution is an operator with dense range. In
particular, we show that the laws of finite-dimensional projections of such
solutions admit smooth densities with respect to Lebesgue measure. The main
idea is to develop a robust pathwise solution theory for such SPDEs using rough
paths theory, which then allows us to use a pathwise version of Norris's lemma
to work directly on the Malliavin matrix, instead of the "reduced Malliavin
matrix" which is not available in this context.
  On our way of proving this result, we develop some new tools for the theory
of rough paths like a rough Fubini theorem and a deterministic mild It\^o
formula for rough PDEs.
