In M-estimation problems involving estimands in Banach spaces, the M-estimators, when appropriately centred and normed, are shown to converge weakly to maximizers of Gaussian processes under rather general conditions. The conventional bootstrap method fails in general to consistently estimate the limit law. We show that the m out of n bootstrap, on the other hand, is weakly consistent under conditions similar to those required for weak convergence of the M-estimators. Strong consistency is also proved under more stringent conditions. Examples of applications are given to illustrate the generality of our results.