The QR-algorithm is one of the most important algorithms in linear algebra. Its several variants make feasible the computation of the eigenvalues and eigenvectors of a numerical real or complex matrix, even when the dimensions of the matrix are enormous. The first adaptation of the QR-algorithm to local fields was given by the first author in 2019. However, in this version the rate of convergence is only linear and in some cases the decomposition into invariant subspaces is incomplete. We present a refinement of this algorithm with a super-linear convergence rate in many cases.