Phys. Rev. D 109, 112006 (2024) We search for the rare decay $B^{+}\rightarrow K^{+}\nu\bar{\nu}$ in a $362\ \rm{fb}^{-1}$ sample of electron-positron collisions at the $\Upsilon(4S)$ resonance collected with the Belle II detector at the SuperKEKB collider. We use the inclusive properties of the accompanying $B$ meson in $\Upsilon(4S) \to B\kern 0.18em\overline{\kern -0.18em B}{}$ 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 $B^{+}\rightarrow K^{+}\nu\bar{\nu}$ branching fraction of $\left2.7\pm 0.5(\mathrm{stat})\pm 0.5(\mathrm{syst})\right \times 10^{-5}$ and $\left1.1^{+0.9}_{-0.8}(\mathrm{stat}){}^{+0.8}_{-0.5}(\mathrm{syst})\right \times 10^{-5}$, respectively. Combining the results, we determine the branching fraction of the decay $B^{+}\rightarrow K^{+}\nu\bar{\nu}$ to be $\left2.3 \pm 0.5(\mathrm{stat})^{+0.5}_{-0.4}(\mathrm{syst})\right\times 10^{-5}$, 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.