In this work, we introduce a memory-efficient method for computing the action of a Hermitian matrix function on a vector. Our method consists of a rational Lanczos algorithm combined with a basis compression procedure based on rational Krylov subspaces that only involve small matrices. The cost of the compression procedure is negligible with respect to the cost of the Lanczos algorithm. This enables us to avoid storing the whole Krylov basis, leading to substantial reductions in memory requirements. This method is particularly effective when the rational Lanczos algorithm needs a significant number of iterations to converge and each iteration involves a low computational effort. This scenario often occurs when polynomial Lanczos as well as extended and shift-and-invert Lanczos are employed. Theoretical results prove that, for a wide variety of functions, the proposed algorithm differs from rational Lanczos by an error term that is usually negligible. The algorithm is compared with other low-memory Krylov methods from the literature on a variety of test problems, showing competitive performance.