We have studied the transverse-momentum (p(T)) dependence of the inclusive J/psi production in p-Pb collisions at root s(NN) = 5.02 TeV, in three center-of-mass rapidity (y(cms)) regions, down to zero p(T). Results in the forward and backward rapidity ranges (2.03 <y(cms) <3.53 and -4.46 <y(cms) <-2.96) are obtained by studying the J/psi decay to mu(+)mu(-), while the mid-rapidity region (-1.37 <y(cms) <0.43) is investigated by measuring the e(+)e(-) decay channel. The p(T) dependence of the J/psi production cross section and nuclear modification factor are presented for each of the rapidity intervals, as well as the J/psi mean p(T) values. Forward and mid-rapidity results show a suppression of the J/psi yield, with respect to pp collisions, which decreases with increasing p(T). At backward rapidity no significant J/psi suppression is observed. Theoretical models including a combination of cold nuclear matter effects such as shadowing and partonic energy loss, are in fair agreement with the data, except