We show partial smoothing properties of the transition semigroup $(P_t)$ associated to the linear stochastic wave equation driven by a cylindrical Wiener noise on a separable Hilbert space. These results allow to study related vector-valued infinite-dimensional PDEs in spaces of functions which are H\"older continuous along special directions and interpolation results involving spaces of H\"older continuous functions along special directions. As an application we prove strong uniqueness for semilinear stochastic wave equations involving nonlinearities of H\"older type.To get pathwise uniqueness we use an approach based on backward stochastic differential equations, different from the so-called Zvonkin transformation or It\^o-Tanaka trick. The talk is based on joint work with E. Priola.