Abstract: A search is presented for massive long-lived particles in events featuring at least one displaced vertex and at least one displaced muon, using proton-proton collision data collected by the ATLAS detector at the Large Hadron Collider from 2022 to 2024 at a centre-of-mass energy of 13.6 TeV. The data sample corresponds to an integrated luminosity of 164 fb(-1). The analysis targets scenarios in which long-lived particles decay inside the ATLAS inner detector, resulting in a topology of at least one massive, displaced vertex (DV) with multiple associated tracks, and at least one muon with a large transverse impact parameter relative to the primary interaction point. The muon is not required to be associated with the DV. Two signal regions are defined by the transverse distance of the reconstructed DV from the interaction point. Background contributions are estimated by using fully data-driven techniques. No significant excess above the expected background is observed. Upper limits at 95% confidence level are set on the visible cross-section and on the production cross-sections of several benchmark models of R-parity-violating supersymmetry.