The decay (B) over bar (0)(s) -> psi(2S)K+pi(-) is observed using a data set corresponding to an integrated luminosity of 3.0 fb(-1) collected by the LHCb experiment in pp collisions at centre-of-mass energies of 7 and 8 TeV. The branching fraction relative to the B-0 -> psi(2S)K+pi(-) decay mode is measured to be B((B) over bar (0)(s) -> psi(2S)K+pi(-))/B(B-0 -> psi(2S)K+pi(-)) = 5.38 +/- 0.36 (stat) +/- 0.22 (syst) +/- 0.31 (f(s)/f(d)) %, where f(s)/f(d) indicates the uncertainty due to the ratio of probabilities for a b quark to hadronise into a B-s(0) or B-0 meson. Using an amplitude analysis, the fraction of decays proceeding via an intermediate K*(892)(0) meson is measured to be 0.645 +/- 0.049 (stat) +/- 0.049 (syst) and its longitudinal polarisation fraction is 0.524 +/- 0.056 (stat) +/- 0.029 (syst). The relative branching fraction for this component is determined to be B((B) over bar (0)(s) -> psi(2S)K*(892)(0))/B(B-0 -> psi(2S)K*(892)(0)) = 5.58 +/- 0.57 (stat) +/- 0.40 (syst) +/- 0.32 (f(s)/f(d)) %. In addition, the mass splitting between the B-s(0) and B-0 mesons is measured as M(B-s(0)) - M(B-0) = 87.45 +/- 0.44 (stat) +/- 0.09 (syst) MeV/c(2). (C) 2015 CERN for the benefit of the LHCb Collaboration. Published by Elsevier B.V. This is an open access article under the CC BY license