A search for a narrow Z ′ gauge boson with a mass between 5 and 70 GeV resulting from an L μ −L τ 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 L μ −L τ 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.