  We consider a class of semilinear Volterra type stochastic evolution equation
driven by multiplicative Gaussian noise. The memory kernel, not necessarily
analytic, is such that the deterministic linear equation exhibits a parabolic
character. Under appropriate Lipschitz-type and linear growth assumptions on
the nonlinear terms we show that the unique mild solution is mean-$p$ H\"older
continuous with values in an appropriate Sobolev space depending on the kernel
and the data. In particular, we obtain pathwise space-time (Sobolev-H\"older)
regularity of the solution together with a maximal type bound on the spatial
Sobolev norm. As one of the main technical tools we establish a smoothing
property of the derivative of the deterministic evolution operator family.
