  We searched for proton decay via $p\to\mu^+K^0$ in 0.37\,Mton$\cdot$years of
data collected between 1996 and 2018 from the Super-Kamiokande water Cherenkov
experiment. The selection criteria were defined separately for $K^0_S$ and
$K^0_L$ channels. No significant event excess has been observed. As a result of
this analysis, which extends the previous search by an additional
0.2\,Mton$\cdot$years of exposure and uses an improved event reconstruction, we
set a lower limit of $3.6\times10^{33}$ years on the proton lifetime.
