  We establish quantitative second-order Sobolev regularity for functions
having a $2$-integrable $p$-Laplacian in bounded RCD spaces, with $p$ in a
suitable range. In the finite-dimensional case, we also obtain Lipschitz
regularity under the assumption that $p$-Laplacian is sufficiently integrable.
Our results cover both $p$-Laplacian eigenfunctions and $p$-harmonic functions
having relatively compact level sets.
