Differential cross sections for the Drell-Yan process, including Z boson production, using the dimuon decay channel are measured in proton-lead (pPb) collisions at a nucleon-nucleon centre-of-mass energy of 8.16 TeV. A data sample recorded with the CMS detector at the LHC is used, corresponding to an integrated luminosity of 173 nb−1. The differential cross section as a function of the dimuon mass is measured in the range 15–600 GeV, for the first time in proton-nucleus collisions. It is also reported as a function of dimuon rapidity over the mass ranges 15–60 GeV and 60–120 GeV, and ratios for the p-going over the Pb-going beam directions are built. In both mass ranges, the differential cross sections as functions of the dimuon transverse momentum pT and of a geometric variable ϕ* are measured, where ϕ* highly correlates with pT but is determined with higher precision. In the Z mass region, the rapidity dependence of the data indicate a modification of the distribution of partons within a lead nucleus as compared to the proton case. The data are more precise than predictions based upon current models of parton distributions.