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 $B_s^0$ 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 $m_1 = 6063.5 pm 1.2 text{ (stat)} pm 0.8text{ (syst) MeV},$ $Gamma_1 = 26 pm 4 text{ (stat)} pm 4text{ (syst) MeV},$ $m_2 = 6114 pm 3 text{ (stat)} pm 5text{ (syst) MeV},$ $Gamma_2 = 66 pm 18 text{ (stat)} pm 21text{ (syst) MeV}.$ Alternative values assuming a decay through $B^{*+}K^-$, with a missing photon from the $B^{*+} rightarrow B^+gamma$ 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 $B_{s2}^{*0}$ state is determined to be $0.87 pm 0.15 text{ (stat)} pm 0.19 text{ (syst)}$.
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