A bstract A measurement of inclusive, prompt, and non-prompt J/ ψ production in p-Pb collisions at a nucleon-nucleon centre-of-mass energy s NN = 5 . 02 TeV is presented. The inclusive J/ ψ mesons are reconstructed in the dielectron decay channel at midrapidity down to a transverse momentum p T = 0. The inclusive J/ ψ nuclear modification factor R pPb is calculated by comparing the new results in p-Pb collisions to a recently measured proton-proton reference at the same centre-of-mass energy. Non-prompt J/ ψ mesons, which originate from the decay of beauty hadrons, are separated from promptly produced J/ ψ on a statistical basis for p T larger than 1.0 GeV/ c . These results are based on the data sample collected by the ALICE detector during the 2016 LHC p-Pb run, corresponding to an integrated luminosity L int = 292 ± 11 μ b − 1 , which is six times larger than the previous publications. The total uncertainty on the p T -integrated inclusive J/ ψ and non-prompt J/ ψ cross section are reduced by a factor 1.7 and 2.2, respectively. The measured cross sections and R pPb are compared with theoretical models that include various combinations of cold nuclear matter effects. From the non-prompt J/ ψ production cross section, the b b ¯ production cross section at midrapidity, d σ b b ¯ / d y , and the total cross section extrapolated over full phase space, σ b b ¯ , are derived.