The production of prompt D+ and D-s(+) mesons is studied in proton-lead collisions at a centre-of-mass energy of root s(NN) = 5.02TeV. The data sample corresponding to an integrated luminosity of (1.58 +/- 0.02)nb(-1) is collected by the LHCb experiment at the LHC. The differential production cross-sections are measured using D+ and D-s(+) candidates with transverse momentum in the range of 0 < p(T) < 14 GeV/c and rapidities in the ranges of 1.5 < y* < 4.0 and -5.0 < y* < -2.5 in the nucleon-nucleon centre-of-mass system. For both particles, the nuclear modification factor and the forward-backward production ratio are determined. These results are compared with theoretical models that include initial-state nuclear effects. In addition, measurements of the cross-section ratios between D+, D-s(+) and D-0 mesons are presented, providing a baseline for studying the charm hadronization in lead-lead collisions at LHC energies.