This paper presents a first measurement of the cross-section for the charged-current Drell-Yan process pp → W$^{±}$ → ℓ$^{±}$ν above the resonance region, where ℓ is an electron or muon. The measurement is performed for transverse masses, $ {m}_{\textrm{T}}^{\textrm{W}} $, between 200 GeV and 5000 GeV, using a sample of 140 fb$^{−1}$ of pp collision data at a centre-of-mass energy of $ \sqrt{s} $ = 13 TeV collected by the ATLAS detector at the LHC during 2015–2018. The data are presented single differentially in transverse mass and double differentially in transverse mass and absolute lepton pseudorapidity. A test of lepton flavour universality shows no significant deviations from the Standard Model. The electron and muon channel measurements are combined to achieve a total experimental precision of 3% at low $ {m}_{\textrm{T}}^{\textrm{W}} $. The single- and double differential W-boson charge asymmetries are evaluated from the measurements. A comparison to next-to-next-to-leading-order perturbative QCD predictions using several recent parton distribution functions and including next-to-leading-order electroweak effects indicates the potential of the data to constrain parton distribution functions. The data are also used to constrain four fermion operators in the Standard Model Effective Field Theory formalism, in particular the lepton-quark operator Wilson coefficient $ {c}_{\ell q}^{(3)} $.[graphic not available: see fulltext]