  The NA62 experiment at the CERN SPS reports a study of a sample of $4
\times10^{9}$ tagged $\pi^0$ mesons from $K^+ \to \pi^+ \pi^0 (\gamma)$,
searching for the decay of the $\pi^0$ to invisible particles. No signal is
observed in excess of the expected background fluctuations. An upper limit of
$4.4 \times10^{-9}$ is set on the branching ratio at 90% confidence level,
improving on previous results by a factor of 60. This result can also be
interpreted as a model-independent upper limit on the branching ratio for the
decay $K^+ \to \pi^+ X$, where $X$ is a particle escaping detection with mass
in the range 0.110-0.155 GeV$/c^2$ and rest lifetime greater than 100 ps.
Model-dependent upper limits are obtained assuming $X$ to be an axion-like
particle with dominant fermion couplings or a dark scalar mixing with the
Standard Model Higgs boson.
