  We prove $p$-adic versions of a classical result in arithmetic geometry
stating that an irreducible subvariety of an abelian variety with dense torsion
has to be the translate of a subgroup by a torsion point. We do so in the
context of certain rigid analytic spaces and formal groups over a $p$-adic
field $K$ or its ring of integers $R$, respectively. In particular, we show
that the rigidity results for algebraic functions underlying the so-called
Manin-Mumford Conjecture generalize to suitable $p$-adic analytic functions. In
the formal setting, this approach leads us to uncover purely $p$-adic
Manin-Mumford type results for formal groups not coming from abelian schemes.
Moreover, we observe that a version of the Tate-Voloch Conjecture holds in the
$p$-adic setting: torsion points either lie squarely on a subscheme or are
uniformly bounded away from it in the $p$-adic distance.
