A structure is observed in the $B+K-$ mass spectrum in a sample of proton--proton collisions at centre-of-mass energies of 7, 8, and 13 TeV, collected with the LHCb detector and corresponding to a total integrated luminosity of 9 fb$-1$. The structure is interpreted as the result of overlapping excited $Bs0$ states. With high significance, a two-peak hypothesis provides a better description of the data than a single resonance. Under this hypothesis the masses and widths of the two states, assuming they decay directly to $B+K-$, are determined to be $m1=6063.5\pm 1.2 (stat)\pm 0.8 (syst)\; MeV,$ $\Gamma 1=26\pm 4 (stat)\pm 4 (syst)\; MeV,$ $m2=6114\pm 3 (stat)\pm 5 (syst)\; MeV,$ $\Gamma 2=66\pm 18 (stat)\pm 21 (syst)\; MeV.$ Alternative values assuming a decay through $B\ast +K-$, with a missing photon from the $B\ast +\to B+\gamma$ decay, which are shifted by approximately 45 MeV are also determined. The possibility of a single state decaying in both channels is also considered. The ratio of the total production cross-section times branching fraction of the new states relative to the previously observed $Bs2\ast 0$ state is determined to be $0.87\pm 0.15 (stat)\pm 0.19 (syst)$.