In order to rigorously define maximum-a-posteriori estimators for nonparametric Bayesian inverse problems for general Banach space valued parameters, we derive and prove certain previously postulated but unproven bounds on small ball probabilities. This allows us to prove existence of MAP estimators in the Banach space setting under very mild assumptions on the loglikelihood. As a similar statement so far (as far as the author is aware) only existed in the Hilbert space setting, this closes an important gap in the literature.