  We study the optimal control of path-dependent McKean-Vlasov equations valued
in Hilbert spaces motivated by non Markovian mean-field models driven by
stochastic PDEs. We first establish the well-posedness of the state equation,
and then we prove the dynamic programming principle (DPP) in such a general
framework. The crucial law invariance property of the value function V is
rigorously obtained, which means that V can be viewed as a function on the
Wasserstein space of probability measures on the set of continuous functions
valued in Hilbert space. We then define a notion of pathwise measure
derivative, which extends the Wasserstein derivative due to Lions [41], and
prove a related functional It{\^o} formula in the spirit of Dupire [24] and Wu
and Zhang [51]. The Master Bellman equation is derived from the DPP by means of
a suitable notion of viscosity solution. We provide different formulations and
simplifications of such a Bellman equation notably in the special case when
there is no dependence on the law of the control.
