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 σ_{b b ¯}/ d y, and the total cross section extrapolated over full phase space, σ_{b b ¯}, 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 R_pPb, and compared to models to study possible nuclear modifications of the production induced by cold nuclear matter effects. R_pPb is found to be smaller than unity at low p_T for both J / ψ coming from beauty hadron decays and prompt J / ψ.