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. (read more)