Muon-induced neutrons can lead to potentially irreducible backgrounds in rare event search experiments. We have investigated the implication of laboratory depth on the muon-induced background in a future dark matter experiment capable of reaching the so-called neutrino floor. Our simulation study focused on a xenon-based detector with 70 tonnes of active mass, surrounded by additional veto systems plus a water shield. Two locations at the Boulby Underground Laboratory (UK) were analysed as examples: an experimental cavern in salt at a depth of 2850 m w. e. (similar to the location of the existing laboratory), and a deeper laboratory located in polyhalite rock at a depth of 3575 m w. e. Our results show that no cosmogenic background events are likely to survive standard analysis cuts for 10 years of operation at either location. The largest background component we identified comes from beta-delayed neutron emission from