The decay $Lambda_b^0 to eta_c(1S) p K^-$ is observed for the first time using a data sample of proton-proton collisions, corresponding to an integrated luminosity of 5.5 $fb^{-1}$, collected with the LHCb experiment at a center-of-mass energy of 13 TeV. The branching fraction of the decay is measured, using the $Lambda_b^0 to J/psi p K^-$ decay as a normalization mode, to be $mathcal{B}(Lambda_b^0 to eta_c(1S) p K^-)=(1.06pm0.16pm0.06^{+0.22}_{-0.19})times10^{-4}$, 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 observed and an upper limit of begin{equation*} frac{mathcal{B}(Lambda_b^0 to P_c(4312)^+ K^-)times mathcal{B}(P_c(4312)^+ to eta_c(1S) p)}{mathcal{B}(Lambda_b^0 to eta_c(1S) p K^-)} < 0.24 end{equation*} is obtained at the 95% confidence level.
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