The decay Lambda(0)(b) -> eta(c) (1S)pK(- )is observed for the first time using a data sample of proton-proton collisions, corresponding to an integrated luminosity of 5.5 fb I, collected with the LHCb experiment at a center-of-mass energy of 13 TeV. The branching fraction of the decay is measured, using the Lambda(0)(b) -> J/psi pK(-) decay as a normalization mode, to be B(Lambda(0)(b) -> eta(c) (1S)pK(-)) = (1.06 +/- 0.16 +/- 0.06(-019)(+0.22)) x 10(-4), where the quoted uncertainties are statistical, systematic and due to external inputs, respectively. A study of the eta(c)(1S)p mass spectrum is performed to search for the P-c(4312)(+) pentaquark state. No evidence is B(Lambda(0)(b) -> P-c(4312)K-+(-))xB(P-c(4312)(+)-> eta(c)(1S)p)/B(Lambda(0)(b) -> eta(c) (1S)pK(-)) < 0.24( ) observed and an upper limit of < 0.24 is obtained at the 95% confidence level.