The associated production of a Higgs boson with a W or Z boson decaying into leptons and where the Higgs boson decays to a bb¯ pair is measured in the high vector-boson transverse momentum regime, above 250 GeV, with the ATLAS detector. The analysed data, corresponding to an integrated luminosity of 139 fb⁻¹, were collected in proton–proton collisions at the Large Hadron Collider between 2015 and 2018 at a centre-of-mass energy of s=13 TeV. The measured signal strength, defined as the ratio of the measured signal yield to that predicted by the Standard Model, is 0.72_−0.36 ^+0.39 corresponding to an observed (expected) significance of 2.1 (2.7) standard deviations. Cross-sections of associated production of a Higgs boson decaying into b quark pairs with a W or Z gauge boson, decaying into leptons, are measured in two exclusive vector boson transverse momentum regions, 250–400 GeV and above 400 GeV, and interpreted as constraints on anomalous couplings in the framework of a Standard Model effective field theory.