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We search for the rare decay$B^{+}\to K^{+}\nu \overline{\nu }$in a$362{\mathrm{fb}}^{-1}$sample of electron-positron collisions at the$\mathrm{\Upsilon }(4S)$resonance collected with the Belle II detector at the SuperKEKB collider. We use the inclusive properties of the accompanying$B$meson in$\mathrm{\Upsilon }(4S)\to B\overline{B}$events to suppress background from other decays of the signal$B$candidate and light-quark pair production. We validate the measurement with an auxiliary analysis based on a conventional hadronic reconstruction of the accompanying$B$meson. For background suppression, we exploit distinct signal features using machine learning methods tuned with simulated data. The signal-reconstruction efficiency and background suppression are validated through various control channels. The branching fraction is extracted in a maximum likelihood fit. Our inclusive and hadronic analyses yield consistent results for the$B^{+}\to K^{+}\nu \overline{\nu }$branching fraction of$[2.7\pm 0.5(\mathrm{stat})\pm 0.5(\mathrm{syst})]\times {10}^{-5}$and$[{1.1}_{-0.8}^{+0.9}(\mathrm{stat}{)}_{-0.5}^{+0.8}(\mathrm{syst})]\times {10}^{-5}$, respectively. Combining the results, we determine the branching fraction of the decay$B^{+}\to K^{+}\nu \overline{\nu }$to be$[2.3\pm 0.5(\mathrm{stat}{)}_{-0.4}^{+0.5}(\mathrm{syst})]\times {10}^{-5}$, providing the first evidence for this decay at 3.5 standard deviations. The combined result is 2.7 standard deviations above the standard model expectation.