The $\Xi_c^+ K^-$ mass spectrum is studied with a sample of $pp$ collision data corresponding to an integrated luminosity of 3.3 fb$^{-1}$, collected by the LHCb experiment. The $\Xi_c^+$ is reconstructed in the decay mode $p K^- \pi^+$. Five new, narrow excited $\Omega_c^0$ states are observed: the $\Omega_c(3000)^0$, $\Omega_c(3050)^0$, $\Omega_c(3066)^0$, $\Omega_c(3090)^0$, and $\Omega_c(3119)^0$. Measurements of their masses and widths are reported.