  For a class of non-autonomous parabolic stochastic partial differential
equations defined on a bounded open subset $D\subset \mathbb {R}^d$ and driven
by an $L^2(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$,
a new result on existence and uniqueness of a mild solution is established.
Compared to the existing results, the uniqueness in a fully nonlinear case is
shown, not assuming the coefficient in front of the noise to be affine.
Additionally, the existence of moments for the solution is established.
