Inclusive ψ(2S) production is measured in p-Pb collisions at the centre-of-mass energy per nucleon-nucleon pair sNN = 8.16 TeV, using the ALICE detector at the CERN LHC. The production of ψ(2S) is studied at forward (2.03 < y_cms< 3.53) and backward (−4.46 < y_cms< −2.96) centre-of-mass rapidity and for transverse momentum p_T< 12 GeV/c via the decay to muon pairs. In this paper, we report the integrated as well as the y_cms- and p_T-differential inclusive production cross sections. Nuclear effects on ψ(2S) production are studied via the determination of the nuclear modification factor that shows a strong suppression at both forward and backward centre-of-mass rapidities. Comparisons with corresponding results for inclusive J/ψ show a similar suppression for the two states at forward rapidity (p-going direction), but a stronger suppression for ψ(2S) at backward rapidity (Pb-going direction). As a function of p_T, no clear dependence of the nuclear modification factor is found. The relative size of nuclear effects on ψ(2S) production compared to J/ψ is also studied via the double ratio of production cross sections [σ_ψ(2S)/σ_J/ψ]pPb/[σ_ψ(2S)/σ_J/ψ]_pp between p-Pb and pp collisions. The results are compared with theoretical models that include various effects related to the initial and final state of the collision system and also with previous measurements at sNN = 5.02 TeV. [Figure not available: see fulltext.]