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The production of $\psi(2S)$ mesons in proton-lead collisions at acentre-of-mass energy per nucleon pair of $\sqrt{s_{NN}}=8.16$ TeV is studiedwith the LHCb detector using data corresponding to an integrated luminosity of34 nb$^{-1}$. The prompt and nonprompt $\psi(2S)$ production cross-sections andthe ratio of the $\psi(2S)$ to $J/\psi$ cross-section are measured as afunction of the meson transverse momentum and rapidity in the nucleon-nucleoncentre-of-mass frame, together with forward-to-backward ratios and nuclearmodification factors. The production of prompt $\psi(2S)$ is observed to bemore suppressed compared to $pp$ collisions than the prompt $J/\psi$production, while the nonprompt productions have similar suppression factors.

Prompt and nonprompt $\psi(2S)$ production in $p$Pb collisions at $\sqrt{s_{NN}}=8.16$ TeV