We report measurements of the production of prompt D⁰, D⁺, D^*+ and D_s ⁺ mesons in Pb–Pb collisions at the centre-of-mass energy per nucleon-nucleon pair sNN=5.02 TeV, in the centrality classes 0–10%, 30–50% and 60–80%. The D-meson production yields are measured at mid-rapidity (|y| < 0.5) as a function of transverse momentum (p_T). The p_T intervals covered in central collisions are: 1 < p_T< 50 GeV/c for D⁰, 2 < p_T< 50GeV/c for D⁺, 3 < p_T< 50GeV/c for D^*+, and 4 < p_T< 16GeV/c for D_s ⁺ mesons. The nuclear modification factors (R_AA) for non-strange D mesons (D⁰, D⁺, D^*+) show minimum values of about 0.2 for p_T = 6–10 GeV/c in the most central collisions and are compatible within uncertainties with those measured at sNN=2.76 TeV. For D_s ⁺ mesons, the values of R_AA are larger than those of non-strange D mesons, but compatible within uncertainties. In central collisions the average R_AA of non-strange D mesons is compatible with that of charged particles for p_T> 8 GeV/c, while it is larger at lower p_T. The nuclear modification factors for strange and non-strange D mesons are also compared to theoretical models with different implementations of in-medium energy loss.[Figure not available: see fulltext.]