  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 $\left[2.7\pm 0.5(\mathrm{stat})\pm
0.5(\mathrm{syst})\right] \times 10^{-5}$ and
$\left[1.1^{+0.9}_{-0.8}(\mathrm{stat}){}^{+0.8}_{-0.5}(\mathrm{syst})\right]
\times 10^{-5}$, respectively. Combining the results, we determine the
branching fraction of the decay $B^{+}\rightarrow K^{+}\nu\bar{\nu}$ to be
$\left[2.3 \pm 0.5(\mathrm{stat})^{+0.5}_{-0.4}(\mathrm{syst})\right]\times
10^{-5}$, 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.
