  We study the Bayesian density estimation of data living in the offset of an
unknown submanifold of the Euclidean space. In this perspective, we introduce a
new notion of anisotropic H\"older for the underlying density and obtain
posterior rates that are minimax optimal and adaptive to the regularity of the
density, to the intrinsic dimension of the manifold, and to the size of the
offset, provided that the latter is not too small -- while still allowed to go
to zero. Our Bayesian procedure, based on location-scale mixtures of Gaussians,
appears to be convenient to implement and yields good practical results, even
for quite singular data.
