  We introduce a $p$-adic analogue of the incomplete gamma function. We also
introduce quantities ($m$-values) associated to a function on natural numbers
and prove a new characterization of $p$-adic continuity for functions with
$p$-integral $m$-values. Combinatorial interpretations for the integral values
of the incomplete gamma function and functions with $m$-values zero or one are
obtained, which show that these functions count derangements in generalized
symmetric groups and permutations with restricted cycle lengths.
