This paper describes precision measurements of the transverse momentum $$p_\mathrm {T}^{\ell \ell }$$ ($$\ell =e,\mu $$) and of the angular variable $$\phi ^{*}_{\eta }$$ distributions of Drell–Yan lepton pairs in a mass range of 66–116 GeV. The analysis uses data from 36.1 fb$$^{-1}$$ of proton–proton collisions at a centre-of-mass energy of $$\sqrt{s}=13\,$$TeV collected by the ATLAS experiment at the LHC in 2015 and 2016. Measurements in electron-pair and muon-pair final states are performed in the same fiducial volumes, corrected for detector effects, and combined. Compared to previous measurements in proton–proton collisions at $$\sqrt{s}=7$$ and $$8\,$$TeV, these new measurements probe perturbative QCD at a higher centre-of-mass energ...