Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For $α\in (1/3,1/2)$, we give criteria for when we can approximate Banach space-valued weakly geometric $α$-rough paths by signatures of curves of bounded variation, given some tuning of the Hölder parameter. We show that these criteria are satisfied for weakly geometric rough paths on Hilbert spaces. As an application, we obtain Wong-Zakai type result for function space valued martingales using the notion of (unbounded) rough drivers.