  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 $\mathcal{B}(\Lambda_b^0 \to \eta_c(1S) p
K^-)=(1.06\pm0.16\pm0.06^{+0.22}_{-0.19})\times10^{-4}$, 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.
