We study path-dependent SDEs in Hilbert spaces. By using methods based on contractions in Banach spaces, we prove existence and uniqueness of mild solutions, continuity of mild solutions with respect to perturbations of all the data of the system, Gâteaux differentiability of generic order n of mild solutions with respect to the starting point, continuity of the Gâteaux derivatives with respect to all the data. The analysis is performed for generic spaces of paths that do not necessarily coincide with the space of continuous functions.