Neutral sodium is a minority species in the atmospheres of late-type stars, and line formation in local thermodynamic equilibrium (LTE) is often a poor assumption, in particular for strong lines. We present an extensive grid of non-LTE calculations for several Na I lines in cool stellar atmospheres, including metal-rich and metal-poor dwarfs and giants. For the first time, we constructed a Na model atom that incorporates accurate quantum mechanical calculations for collisional excitation and ionisation by electrons as well as collisional excitation and charge exchange reactions with neutral hydrogen. Similar to Li I, the new rates for hydrogen impact excitation do not affect the statistical equilibrium calculations, while charge exchange reactions have a small but non-negligible influence. The presented LTE and non-LTE curves-of-growth can be interpolated to obtain non-LTE abundances and abundance corrections for arbitrary stellar parameter combinations and line strengths. The typical corrections for weak lines are −0.1... −0.2 dex, whereas saturated lines may overestimate the abundance in LTE by more than 0.5 dex. The non-LTE Na abundances appear very robust with respect to uncertainties in the input collisional data.