  The first full amplitude analysis of $B^+ \to \psi(2S) K^+ \pi^+ \pi^-$
decays is performed using proton-proton collision data corresponding to an
integrated luminosity of $9\,\text{fb}^{-1}$ 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 $X^0 \to \psi(2S)
\rho(770)^0$ decays, broadly resembles the $J/\psi \phi$ spectrum observed in
$B^+ \to J/\psi \phi K^+$ decays. Exotic $\psi(2S) K^+ \pi^-$ resonances are
observed for the first time.
