  We extend Peng's maximum principle for semilinear stochastic partial
differential equations (SPDEs) in one space-dimension with non-convex control
domains and control-dependent diffusion coefficients to the case of general
cost functionals with Nemytskii-type coefficients. Our analysis is based on a
new approach to the characterization of the second order adjoint state as the
solution of a function-valued backward SPDE.
