  We prove $L^p$ bounds for the extensions of standard multilinear
Calder\'on-Zygmund operators to tuples of UMD spaces tied by a natural product
structure. This can, for instance, mean the pointwise product in UMD function
lattices, or the composition of operators in the Schatten-von Neumann subclass
of the algebra of bounded operators on a Hilbert space. We do not require
additional assumptions beyond UMD on each space - in contrast to previous
results, we e.g. show that the Rademacher maximal function property is not
necessary. The obtained generality allows for novel applications. For instance,
we prove new versions of fractional Leibniz rules via our results concerning
the boundedness of multilinear singular integrals in non-commutative $L^p$
spaces. Our proof techniques combine a novel scheme of induction on the
multilinearity index with dyadic-probabilistic techniques in the UMD space
setting.
