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)^-) &= 6315.64pm0.31pm0.07pm0.50 {rm MeV}, m(Omega_b(6330)^-) &= 6330.30pm0.28pm0.07pm0.50 {rm MeV}, m(Omega_b(6340)^-) &= 6339.71pm0.26pm0.05pm0.50 {rm MeV}, m(Omega_b(6350)^-) &= 6349.88pm0.35pm0.05pm0.50 {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.
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