An amplitude analysis of the decay Lambda(0)(b) -> D(0)p pi(-) is performed in the part of the phase space containing resonances in the D(0)p channel. The study is based on a data sample corresponding to an integrated luminosity of 3.0 fb(-1) of pp collisions recorded by the LHCb experiment. The spectrum of excited Lambda(+)(c) states that decay into D(0)p is studied. The masses, widths and quantum numbers of the Lambda(c)(2880)(+) and Lambda(c) (2940)(+) resonances are measured. The constraints on the spin and parity for the Lambda(c)(2940)(+) state are obtained for the first time. A near-threshold enhancement in the D(0)p amplitude is investigated and found to be consistent with a new resonance, denoted the Lambda(c) (2860)(+), of spin 3/2 and positive parity.