The measurements of the production of prompt D , D ⁺ , D ^{∗ +} , and Ds+ mesons in proton–proton (pp) collisions at s=5.02TeV with the ALICE detector at the Large Hadron Collider (LHC) are reported. D mesons were reconstructed at mid-rapidity (| y| < 0.5) via their hadronic decay channels D → K ^- π ⁺ , D ⁺ → K ^- π ⁺ π ⁺ , D ^{∗ +} → D π ⁺ → K ^- π ⁺ π ⁺ , Ds+→ϕπ+→K+K-π+, and their charge conjugates. The production cross sections were measured in the transverse momentum interval 0<pT<36GeV/c for D , 1<pT<36GeV/c for D ⁺ and D ^{∗ +} , and in 2<pT<24GeV/c for Ds+ mesons. Thanks to the higher integrated luminosity, an analysis in finer p _T bins with respect to the previous measurements at s=7TeV was performed, allowing for a more detailed description of the cross-section p _T shape. The measured p _T -differential production cross sections are compared to the results at s=7 TeV and to four different perturbative QCD calculations. Its rapidity dependence is also tested combining the ALICE and LHCb measurements in pp collisions at s=5.02TeV. This measurement will allow for a more accurate determination of the nuclear modification factor in p–Pb and Pb–Pb collisions performed at the same nucleon–nucleon centre-of-mass energy.