The pT-differential production cross sections of the prompt charmed mesons D0, D+, D*+, and D+s and their charge conjugate in the rapidity interval –0.96 < ycms < 0.04 were measured in p–Pb collisions at a center-of-mass energy √sNN = 5.02 TeV with the ALICE detector at the LHC. The nuclear modification factor RpPb, quantifying the D-meson yield in p–Pb collisions relative to the yield in pp collisions scaled by the number of binary nucleon-nucleon collisions, is compatible within the 15%–20% uncertainties with unity in the transverse momentum interval 1 < pT < 24 GeV/c. No significant difference among the RpPb of the four D-meson species is observed. The results are described within uncertainties by theoretical calculations that include initial-state effects. In conclusion, the measurement adds experimental evidence that the modification of the momentum spectrum of D mesons observed in Pb-Pb collisions with respect to pp collisions is due to strong final-state effects induced by hot partonic matter.