A measurement of beauty hadron production at mid-rapidity in proton-lead collisions at a nucleon–nucleon centre-of-mass energy sNN=5.02 TeV is presented. The semi-inclusive decay channel of beauty hadrons into J / ψ is considered, where the J / ψ mesons are reconstructed in the dielectron decay channel at mid-rapidity down to transverse momenta of 1.3 GeV/c. The b b ¯ production cross section at mid-rapidity, d σbb ¯/ d y, and the total cross section extrapolated over full phase space, σbb ¯, are obtained. This measurement is combined with results on inclusive J / ψ production to determine the prompt J / ψ cross sections. The results in p–Pb collisions are then scaled to expectations from pp collisions at the same centre-of-mass energy to derive the nuclear modification factor RpPb, and compared to models to study possible nuclear modifications of the production induced by cold nuclear matter effects. RpPb is found to be smaller than unity at low pT for both J / ψ coming from beauty hadron decays and prompt J / ψ.