Abstract This paper describes precision measurements of the transverse momentum $$p_\mathrm {T}^{\ell \ell }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>p</mml:mi><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>ℓ</mml:mi></mml:mrow></mml:msubsup></mml:math> ( $$\ell =e,\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>=</mml:mo><mml:mi>e</mml:mi><mml:mo>,</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:math> ) and of the angular variable $$\phi ^{*}_{\eta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>ϕ</mml:mi><mml:mi>η</mml:mi><mml:mrow><mml:mrow /><mml:mo>∗</mml:mo></mml:mrow></mml:msubsup></mml:math> distributions of Drell–Yan lepton pairs in a mass range of 66–116 GeV. The analysis uses data from 36.1 fb $$^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow /><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> of proton–proton collisions at a centre-of-mass energy of $$\sqrt{s}=13\,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msqrt><mml:mi>s</mml:mi></mml:msqrt><mml:mo>=</mml:mo><mml:mn>13</mml:mn><mml:mspace /></mml:mrow></mml:math> 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 $$\sqrt{s}=7$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msqrt><mml:mi>s</mml:mi></mml:msqrt><mml:mo>=</mml:mo><mml:mn>7</mml:mn></mml:mrow></mml:math> and $$8\,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>8</mml:mn><mml:mspace /></mml:mrow></mml:math> TeV, these new measurements probe perturbative QCD at a higher centre-of-mass energy with a different composition of initial states. They reach a precision of 0.2 $$\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>%</mml:mo></mml:math> for the normalized spectra at low values of $$p_\mathrm {T}^{\ell \ell }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>p</mml:mi><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mi>ℓ</mml:mi></mml:mrow></mml:msubsup></mml:math> . The data are compared with different QCD predictions, where it is found that predictions based on resummation approaches can describe the full spectrum within uncertainties.