The transverse momentum (p(T)) distribution of primary charged particles is measured in minimum bias (non-single-diffractive) p+Pb collisions at sqrt[s(NN)]=5.02 TeV with the ALICE detector at the LHC. The p(T) spectra measured near central rapidity in the range 0.5<p(T) <20 GeV/c exhibit a weak pseudorapidity dependence. The nuclear modification factor R(pPb) is consistent with unity for p(T) above 2 GeV/c. This measurement indicates that the strong suppression of hadron production at high p(T) observed in Pb+Pb collisions at the LHC is not due to an initial-state effect. The measurement is compared to theoretical calculations.