We perform an extensive suite of N-body simulations of the matter power spectrum, incorporating massive neutrinos in the range Mν= 0.15–0.6 eV, probing the non-linear regime at scales k < 10 h Mpc−1 at z < 3. We extend the widely used halofit approximation to account for the effect of massive neutrinos on the power spectrum. In the strongly non-linear regime, halofit systematically overpredicts the suppression due to the free streaming of the neutrinos. The maximal discrepancy occurs at k∼ 1h Mpc−1, and is at the level of 10 per cent of the total suppression. Most published constraints on neutrino masses based on halofit are not affected, as they rely on data probing the matter power spectrum in the linear or mildly non-linear regime. However, predictions for future galaxy, Lyman α forest and weak lensing surveys extending to more non-linear scales will benefit from the improved approximation to the non-linear matter power spectrum we provide. Our approximation reproduces the induced neutrino suppression over the targeted scales and redshifts significantly better. We test its robustness with regard to changing cosmological parameters and a variety of modelling effects.