A search for a narrow Z' gauge boson with a mass between 5 and 70 GeV resulting from an L-mu - L-tau 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'mu mu -> 4 mu 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'mu mu)B(Z' -> mu mu), 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-mu - L-tau 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. (C) 2019 The Author(s). Published by Elsevier B.V.