A search for a new 𝑍′ gauge boson predicted by 𝐿𝜇−𝐿𝜏 models, based on charged-current Drell–Yan production, 𝑝⁢𝑝→𝑊±(*)→𝑍′⁢𝜇±⁢𝜈→𝜇±⁢𝜇∓⁢𝜇±⁢𝜈, is presented. The data sample used corresponds to an integrated luminosity of 140  fb−1 of proton–proton collisions at √𝑠=13  TeV recorded by the ATLAS detector at the Large Hadron Collider. The search examines a final state of 3⁢𝜇 plus large missing transverse momentum. Upper limits are set on the 𝑍′ production cross section times branching ratio in the mass range of 5–81 GeV. After combining with the previous 𝑍′ search using the neutral-current Drell–Yan production with a 4⁢𝜇 final state, the most stringent exclusion limits to date are achieved in the parameter space of the 𝑍′ coupling strength and mass.