  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
  $m_1 = 6063.5 \pm 1.2 \text{ (stat)} \pm 0.8\text{ (syst) MeV},$
  $\Gamma_1 = 26 \pm 4 \text{ (stat)} \pm 4\text{ (syst) MeV},$
  $m_2 = 6114 \pm 3 \text{ (stat)} \pm 5\text{ (syst) MeV},$
  $\Gamma_2 = 66 \pm 18 \text{ (stat)} \pm 21\text{ (syst) MeV}.$
  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 $0.87 \pm 0.15
\text{ (stat)} \pm 0.19 \text{ (syst)}$.
