  We put forward a general framework for the study of a pathwise central limit
theorem (CLT) and a moderate deviation principle (MDP) for stochastic partial
differential equations perturbed with a small multiplicative linear noise by
means of the theory of rough paths. The CLT can be interpreted as the
convergence to a pathwise derivative of the It\^o-Lyons map. The result follows
by applying a pathwise Malliavin-like calculus for rough paths and from
compactness methods. The convergence in the CLT is quantified by an optimal
speed of convergence. From the exponential equivalence principle and the
knowledge of the speed of convergence, we can derive easily a MDP. In
particular, we do not apply the weak convergence approach usually employed in
this framework. We derive a pathwise CLT and a MDP for the stochastic
Landau-Lifschitz-Gilbert equation in one dimension, for the heat equation and
for a stochastic reaction-diffusion equation. As a further application, we
derive a pathwise convergence to the CLT limit and a corresponding MDP for
equations driven by linear It\^o noise.
