The first full amplitude analysis of B+ -> psi(2S)K+pi(+)pi(-) decays is performed using proton-proton collision data corresponding to an integrated luminosity of 9 fb(-1) recorded with the LHCb detector. The rich K+pi(+)pi(-) spectrum is studied and the branching fractions of the resonant substructure associated with the prominent K-1(1270)(+) contribution are measured. The data cannot be described by conventional strange and charmonium resonances only. An amplitude model with 53 components is developed comprising 11 hidden-charm exotic hadrons. New production mechanisms for charged charmonium-like states are observed. Significant resonant activity with spin-parity J(P) = 1(+) in the psi(2S)pi(+) system is confirmed and a multi-pole structure is demonstrated. The spectral decomposition of the psi(2S)pi(+)pi(-) invariant-mass structure, dominated by X-0 -> psi(2S)rho(770)(0) decays, broadly resembles the J/psi phi spectrum observed in B+ -> J/psi phi K+ decays. Exotic psi(2S)K+pi(-) resonances are observed for the first time.