In this article, we study the binary classification problem with supervised data, in the case where the covariate-to-probability-of-success map is possibly spatially inhomogeneous. We devise nonparametric Bayesian procedures with Besov-Laplace priors, which are prior distributions on function spaces routinely used in imaging and inverse problems in view of their useful edge-preserving and sparsity-promoting properties. Building on a recent line of work in the literature, we investigate the theoretical asymptotic recovery properties of the associated posterior distributions, and show that suitably tuned Besov-Laplace priors lead to minimax-optimal posterior contraction rates as the sample size increases, under the frequentist assumption that the data have been generated by a spatially inhomogeneous ground truth belonging to a Besov space.