Fiducial and differential cross-section measurements of W⁺W⁻ production in association with at least one hadronic jet are presented. These measurements are sensitive to the properties of electroweak-boson self-interactions and provide a test of perturbative quantum chromodynamics and the electroweak theory. The analysis is performed using proton-proton collision data collected at s = 13 TeV with the ATLAS experiment, corresponding to an integrated luminosity of 139 fb⁻¹. Events are selected with exactly one oppositely charged electron-muon pair and at least one hadronic jet with a transverse momentum of p_T> 30 GeV and a pseudorapidity of |η| < 4.5. After subtracting the background contributions and correcting for detector effects, the jet-inclusive W⁺W⁻+ ≥ 1 jet fiducial cross-section and W⁺W⁻+ jets differential cross-sections with respect to several kinematic variables are measured. These measurements include leptonic quantities, such as the lepton transverse momenta and the transverse mass of the W⁺W⁻ system, as well as jet-related observables such as the leading jet transverse momentum and the jet multiplicity. Limits on anomalous triple-gauge-boson couplings are obtained in a phase space where interference between the Standard Model amplitude and the anomalous amplitude is enhanced. [Figure not available: see fulltext.]