The production of J/psi mesons is studied in proton-lead collisions at the centre-of-mass energy per nucleon pair root s(NN) = 8.16 TeV with the LHCb detector at the LHC. The double differential cross-sections of prompt and nonprompt J/psi production are measured as a function of the J/psi transverse momentum and rapidity in the nucleon-nucleon centre-of-mass frame. Forward-to-backward ratios and nuclear modification factors are determined. The results are compared with theoretical calculations based on collinear factorisation using nuclear parton distribution functions, on the colour glass condensate or on coherent energy loss models. (c) 2017 The Author. Published by Elsevier B.V.