A full amplitude analysis of Lambda(0)(b) -> J/psi pi(-) decays is performed with a data sample acquired with the LHCb detector from 7 and 8 TeV pp collisions, corresponding to an integrated luminosity of 3 fb(-1). A significantly better description of the data is achieved when, in addition to the previously observed nucleon excitations N -> p pi(-), either the P-c(43800)(+) and P-c(4450)(+) -> J/psi p states, previously observed in Lambda(0)(b) -> J/psi pK(-) decays, or the Z(c)(4200)(-) -> J/psi pi(-) state, previously reported in B-0 -> J/psi K+pi(-) decays, or all three, are included in the amplitude models. The data support a model containing all three exotic states, with a significance of more than three standard deviations. Within uncertainties, the data are consistent with the P-c(4380)(+) and P-c(4 450)(+) production rates expected from their previous observation taking account of Cabibbo suppression.