A full amplitude analysis of Λ_{b}^{0}→J/ψpπ^{-} 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→pπ^{-}, either the P_{c}(4380)^{+} and P_{c}(4450)^{+}→J/ψp states, previously observed in Λ_{b}^{0}→J/ψpK^{-} decays, or the Z_{c}(4200)^{-}→J/ψπ^{-} state, previously reported in B^{0}→J/ψK^{+}π^{-} 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}(4450)^{+} production rates expected from their previous observation taking account of Cabibbo suppression.