Summary: The fundamental solutions to a large class of pseudo-differential equations that generalize the formal analogy of the diffusion equation in \(\mathbb {R}\) to the groups \(p^{-n}\mathbb {Z}_p/p^{n} \mathbb {Z}_p\) give rise to probability measures on the space of Skorokhod paths on these finite groups. These measures induce probability measures on the Skorokhod space of \(\mathbb {Q}_p\)-valued paths that almost surely take values on finite grids. We study the convergence of these induced measures to their continuum limit, a \(p\)-adic Brownian motion. We additionally prove a Feynman-Kac formula for the matrix-valued propagator associated to a Schrödinger type operator acting on complex vector-valued functions on \(p^{-n}\mathbb {Z}_p/p^{n} \mathbb {Z}_p\) where the potential is a Hermitian matrix-valued multiplication operator.