  A search for the rare decay $K_L \!\to\! \pi^0 \nu \overline{\nu}$ was
performed. With the data collected in 2015, corresponding to $2.2 \times
10^{19}$ protons on target, a single event sensitivity of $( 1.30 \pm 0.01_{\rm
stat} \pm 0.14_{\rm syst} ) \times 10^{-9}$ was achieved and no candidate
events were observed. We set an upper limit of $3.0 \times 10^{-9}$ for the
branching fraction of $K_L \!\to\! \pi^0 \nu \overline{\nu}$ at the 90%
confidence level (C.L.), which improved the previous limit by almost an order
of magnitude. An upper limit for $K_L \!\to\! \pi^0 X^0$ was also set as $2.4
\times 10^{-9}$ at the 90% C.L., where $X^0$ is an invisible boson with a mass
of $135~{\rm MeV}/c^2$.
