A bstract This paper presents a search for a new Z ′ vector gauge boson with the ATLAS experiment at the Large Hadron Collider using pp collision data collected at $$ \sqrt{s} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>s</mml:mi> </mml:msqrt> </mml:math> = 13 TeV, corresponding to an integrated luminosity of 139 fb − 1 . The new gauge boson Z ′ is predicted by L μ − L τ models to address observed phenomena that can not be explained by the Standard Model. The search examines the four-muon (4 μ ) final state, using a deep learning neural network classifier to separate the Z ′ signal from the Standard Model background events. The di-muon invariant masses in the 4 μ events are used to extract the Z ′ resonance signature. No significant excess of events is observed over the predicted background. Upper limits at a 95% confidence level on the Z ′ production cross-section times the decay branching fraction of pp → Z ′ μμ → 4 μ are set from 0.31 to 4.3 fb for the Z ′ mass ranging from 5 to 81 GeV. The corresponding common coupling strengths, g Z ′ , of the Z ′ boson to the second and third generation leptons above 0.003 – 0.2 have been excluded.