The production of ψ(2S) mesons is studied in dimuon final states using proton-lead (pPb) collision data collected by the LHCb detector. The data sample corresponds to an integrated luminosity of 1.6 nb−1. The nucleon-nucleon centre-of-mass energy of the pPb collisions is sNN−−−−√=5sNN=5 TeV. The measurement is performed using ψ(2S) mesons with transverse momentum less than 14 GeV/c and rapidity y in the ranges 1.5 < y < 4.0 and −5.0 < y < −2.5 in the nucleon-nucleon centre-of-mass system. The forward-backward production ratio and the nuclear modification factor are determined for ψ(2S) mesons. Using the production cross-section results of ψ(2S) and J/ψ mesons from b-hadron decays, the bb¯¯bb¯ cross-section in pPb collisions at sNN−−−−√=5sNN=5 TeV is obtained.