  We conduct a model-independent analysis of the distinct signatures of various
generic new physics possibilities in the decay $B \to K \, \nu \,
\overline{\nu}$ by analyzing the branching ratio as well as the missing
mass-square distribution. Considering the final neutrinos to be of the same
flavor with non-zero mass, we discuss the new physics contributions for both
Dirac and Majorana neutrino possibilities. In our study, we utilize the
analytical relations among form factors in semi-leptonic $B \to K$ transitions,
which are consistent with current lattice QCD predictions to a very high
numerical accuracy. We provide constraints on different new physics parameters,
taking into account the recent measurement of $B^+ \to K^+ \, \nu \,
\overline{\nu}$ branching ratio by the Belle-II collaboration. In future, if
the missing mass-square distribution for $B^+ \to K^+ \, \nu \, \overline{\nu}$
decay gets reported by Belle-II with analysis of more events than their present
data set, one can not only investigate possible new physics effects in these
decays, but also probe the Dirac/Majorana nature of the neutrinos using quantum
statistics, since a difference between the two cases is known to exist in the
presence of non-standard neutrino interactions.
