A bstract The production of ψ (2 S ) mesons in proton-lead collisions at a centre-of-mass energy per nucleon pair of $$ \sqrt{s_{\textrm{NN}}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>NN</mml:mi> </mml:msub> </mml:msqrt> </mml:math> = 8 . 16 TeV is studied with the LHCb detector using data corresponding to an integrated luminosity of 34 nb − 1 . The prompt and nonprompt ψ (2 S ) production cross-sections and the ratio of the ψ (2 S ) to J/ψ cross-section are measured as a function of the meson transverse momentum and rapidity in the nucleon-nucleon centre-of-mass frame, together with forward-to-backward ratios and nuclear modification factors. The production of prompt ψ (2 S ) is observed to be more suppressed compared to pp collisions than the prompt J/ψ production, while the nonprompt productions have similar suppression factors.