Phys. Rev. Lett. 124, 082002 (2020) We report four narrow peaks in the $\Xi_b^0K^-$ mass spectrum obtained using $pp$ collisions at center-of-mass energies of 7, 8 and 13 TeV, corresponding to a total integrated luminosity of 9 fb$^{-1}$ recorded by the LHCb experiment. Referring to these states by their mass, the mass values are \begin{align*} m(\Omega_b(6316)^-) &= 6315.64\pm0.31\pm0.07\pm0.50 {\rm MeV}, \\ m(\Omega_b(6330)^-) &= 6330.30\pm0.28\pm0.07\pm0.50 {\rm MeV}, \\ m(\Omega_b(6340)^-) &= 6339.71\pm0.26\pm0.05\pm0.50 {\rm MeV}, \\ m(\Omega_b(6350)^-) &= 6349.88\pm0.35\pm0.05\pm0.50 {\rm MeV}, \end{align*}where the uncertainties are statistical, systematic and the last is due to the knowledge of the $\Xi_b^0$ mass. The natural widths of the three lower mass states are consistent with zero, and the 90% confidence-level upper limits are determined to be ${\Gamma(\Omega_b(6316)^-)<2.8}$ MeV, ${\Gamma(\Omega_b(6330)^-)<3.1}$ MeV and ${\Gamma(\Omega_b(6340)^-)<1.5}$ MeV. The natural width of the $\Omega_b(6350)^-$ peak is $1.4^{+1.0}_{-0.8}\pm0.1$ MeV, which is 2.5$\sigma$ from zero and corresponds to an upper limit of 2.8 MeV. The peaks have local significances ranging from 3.6$\sigma$ to 7.2$\sigma$. After accounting for the look-elsewhere effect, the significances of the $\Omega_b(6316)^-$ and $\Omega_b(6330)^-$ peaks are reduced to 2.1$\sigma$ and 2.6$\sigma$ respectively, while the two higher mass peaks exceed 5$\sigma$. The observed peaks are consistent with expectations for excited $\Omega_b^-$ resonances.