  We establish n-th order Fr\'echet differentiability with respect to the
initial datum of mild solutions to a class of jump-diffusions in Hilbert
spaces. In particular, the coefficients are Lipschitz continuous, but their
derivatives of order higher than one can grow polynomially, and the
(multiplicative) noise sources are a cylindrical Wiener process and a
quasi-left-continuous integer-valued random measure. As preliminary steps, we
prove well-posedness in the mild sense for this class of equations, as well as
first-order G\^ateaux differentiability of their solutions with respect to the
initial datum, extending previous results in several ways. The
differentiability results obtained here are a fundamental step to construct
classical solutions to non-local Kolmogorov equations with sufficiently regular
coefficients by probabilistic means.
