  The Belle II experiment recently observed the decay $B^+ \to K^+ \nu \nu$ for
the first time, with a measured value for the branching ratio of $ (2.3 \pm
0.7) \times 10^{-5}$. This result exhibits a $\sim 3\sigma$ deviation from the
Standard Model (SM) prediction. The observed enhancement with respect to the
Standard Model could indicate the presence of invisible light new physics. In
this paper, we investigate whether this result can be accommodated in a minimal
Higgs portal model, where the SM is extended by a singlet Higgs scalar that
decays invisibly to dark sector states. We find that current and future bounds
on invisible decays of the 125 GeV Higgs boson completely exclude a new scalar
with a mass $\gtrsim 10$ GeV. On the other hand, the Belle II results can be
successfully accommodated if the new scalar is lighter than $B$ mesons but
heavier than kaons. We also investigate the cosmological implications of the
new states and explore the possibility that they are part of an abelian Higgs
extension of the SM. Future Higgs factories are expected to place stringent
bounds on the invisible branching ratio of the 125 GeV Higgs boson, and will be
able to definitively test the region of parameter space favored by the Belle II
results.
