  We introduce the notion of $p$-adic quantum bit ($p$-qubit) in the context of
the $p$-adic quantum mechanics initiated and developed by Volovich and his
followers. In this approach, physics takes place in three-dimensional $p$-adic
space rather than Euclidean space. Based on our prior work describing the
$p$-adic special orthogonal group, we outline a programme to classify its
continuous unitary projective representations, which can be interpreted as a
theory of $p$-adic angular momentum. The $p$-adic quantum bit arises from the
irreducible representations of minimal nontrivial dimension two, of which we
construct examples for all primes $p$.
