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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 suchfunctions 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 theformal multiplicative group. For polynomial functions, this follows from themultiplicative Manin-Mumford conjecture. However we allow for a much widerclass of analytic functions; in particular we establish a rigidity result forformal tori. Moreover, our methods apply to Lubin-Tate formal groups beyondjust the formal multiplicative group and we extend the results to this setting.

An Infinitesimal $p$-adic Multiplicative Manin-Mumford Conjecture