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A search for a narrow Z′ gauge boson with a mass between 5 and 70 GeV resulting from an Lμ−Lτ U(1) local gauge symmetry is reported. Theories that predict such a particle have been proposed as an explanation of various experimental discrepancies, including the lack of a dark matter signal in direct-detection experiments, tension in the measurement of the anomalous magnetic moment of the muon, and reports of possible lepton flavor universality violation in B meson decays. A data sample of proton-proton collisions at a center-of-mass energy of 13 TeV is used, corresponding to an integrated luminosity of 77.3 fb−1 recorded in 2016 and 2017 by the CMS detector at the LHC. Events containing four muons with an invariant mass near the standard model Z boson mass are analyzed, and the selection is further optimized to be sensitive to the events that may contain Z→Z′μμ→4μ decays. The event yields are consistent with the standard model predictions. Upper limits of 10−8-10−7 at 95% confidence level are set on the product of branching fractions B(Z→Z′μμ)B(Z′→μμ), depending on the Z′ mass, which excludes a Z′ boson coupling strength to muons above 0.004-0.3. These are the first dedicated limits on Lμ−Lτ models at the LHC and result in a significant increase in the excluded model parameter space. The results of this search may also be used to constrain the coupling strength of any light Z′ gauge boson to muons.