The ALICE Collaboration has studied the inclusive production of the charmonium state ψ(2S) in proton-lead (p-Pb) collisions at the nucleon-nucleon centre of mass energy sNN−−−√ = 5.02 TeV at the CERN LHC. The measurement was performed at forward (2.03 < y cms < 3.53) and backward (−4.46 < y cms < −2.96) centre of mass rapidities, studying the decays into muon pairs. In this paper, we present the inclusive production cross sections σ ψ(2S), both integrated and as a function of the transverse momentum p T, for the two y cms domains. The results are compared to those obtained for the 1S vector state (J/ψ), by showing the ratios between the production cross sections, as well as the double ratios [σ ψ(2S) /σ J/ψ ]pPb /[σ ψ(2S) /σ J/ψ ]pp between p-Pb and proton-proton collisions. Finally, the nuclear modification factor for inclusive ψ(2S) is evaluated and compared to the measurement of the same quantity for J/ψ and to theoretical models including parton shadowing and coherent energy loss mechanisms. The results show a significantly larger suppression of the ψ(2S) compared to that measured for J/ψ and to models. These observations represent a clear indication for sizeable final state effects on ψ(2S) production.