  We consider semilinear stochastic evolution equations on Hilbert spaces with
multiplicative Wiener noise and linear drift term of the type $A + \varepsilon
G$, with $A$ and $G$ maximal monotone operators and $\varepsilon$ a "small"
parameter, and study the differentiability of mild solutions with respect to
$\varepsilon$. The operator $G$ can be a singular perturbation of $A$, in the
sense that its domain can be strictly contained in the domain of $A$.
