The measurement of prompt D-meson production as a function of multiplicity in p-Pb collisions at sNN=5.02 TeV with the ALICE detector at the LHC is reported. D0, D+ and D∗+ mesons are reconstructed via their hadronic decay channels in the centre-of-mass rapidity range −0.96 < ycms< 0.04 and transverse momentum interval 1<pT<24 GeV/c. The multiplicity dependence of D-meson production is examined by either comparing yields in p-Pb collisions in different event classes, selected based on the multiplicity of produced particles or zero-degree energy, with those in pp collisions, scaled by the number of binary nucleon-nucleon collisions (nuclear modification factor); as well as by evaluating the per-event yields in p-Pb collisions in different multiplicity intervals normalised to the multiplicity-integrated ones (relative yields). The nuclear modification factors for D0, D+ and D∗+ are consistent with one another. The D-meson nuclear modification factors as a function of the zero-degree energy are consistent with unity within uncertainties in the measured pT regions and event classes. The relative D-meson yields, calculated in various pT intervals, increase as a function of the charged-particle multiplicity. The results are compared with the equivalent pp measurements at s=7 TeV as well as with EPOS 3 calculations.[Figure not available: see fulltext.]