A bstract The first full amplitude analysis of B + → ψ (2 S ) K + π + π − 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 + π + π − 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 ψ (2 S ) π + system is confirmed and a multi-pole structure is demonstrated. The spectral decomposition of the ψ (2 S ) π + π − invariant-mass structure, dominated by X 0 → ψ (2 S ) ρ (770) 0 decays, broadly resembles the J / ψϕ spectrum observed in B + → J / ψϕK + decays. Exotic ψ (2 S ) K + π − resonances are observed for the first time.