  Our results concern analytic functions on the open unit $p$-adic poly-disc in
$\mathbb{C}^n_p$ centered at the multiplicative unit and we prove that such
functions only vanish at finitely many $n$-tuples of roots of unity
$(\zeta_1-1,\ldots,\zeta_n-1)$ unless they vanish along a translate of the
formal multiplicative group. For polynomial functions, this follows from the
multiplicative Manin-Mumford conjecture. However we allow for a much wider
class of analytic functions; in particular we establish a rigidity result for
formal tori. Moreover, our methods apply to Lubin-Tate formal groups beyond
just the formal multiplicative group and we extend the results to this setting.
