The first search for the B_{s}^{0}→K^{-}π^{+}γ decay in the range 796<m(K^{-}π^{+})<1800 MeV/c^{2} is performed using data from proton-proton collisions collected by the LHCb experiment at center-of-mass energies of 7, 8, and 13 TeV, corresponding to an integrated luminosity of 9 fb^{-1}. The photons are reconstructed through their conversion into an electron-positron pair, which significantly improves the mass resolution of the reconstructed decays with respect to decays with an unconverted photon. A signal excess with a significance of 3.5 standard deviations is measured, constituting the first experimental evidence for this decay. In the range 796<m(K^{-}π^{+})<996 MeV/c^{2}, the ratio R between the branching fractions of the signal decay and the favored B[over ¯]^{0}→K^{-}π^{+}γ decay is measured to be R=(3.7±1.2±0.4)×10^{-2} where the first uncertainty is statistical and the second is systematic. This measurement is consistent with the value predicted in the Standard Model. In the range 996<m(K^{-}π^{+})<1800 MeV/c^{2}, the ratio R=(0.2±2.7±1.3)×10^{-2} is measured.