We study nonparametric Bayesian binary classification, in the case where the unknown probability response function is possibly spatially inhomogeneous, for example, being generally flat across the domain but presenting localized sharp variations. We consider a hierarchical procedure based on the Besov-Laplace priors from the inverse problems and imaging literature, with a carefully tuned hyper-prior on the regularity parameter. We show that the resulting posterior distribution concentrates towards the ground truth at optimal rate, automatically adapting to the unknown regularity. To implement posterior inference in practice, we devise an efficient Markov chain Monte Carlo (MCMC) algorithm based on recent ad-hoc dimension-robust methods for Besov-Laplace priors. We then test the considered approach in extensive numerical simulations, where we obtain a solid corroboration of the theoretical results.