In 1970, E. Bombieri and S. Lang used analytic results of P. Lelong to establish a Schwarz lemma for a well distributed set of points in $\mathbb {C}^n$. Their result was extended to an interpolation lemma, first by D.W. Masser in the case of polynomials, then by M. Waldschmidt for analytic functions. J.-C. Moreau gave an analog of it over the real numbers and P. Robba in the $p$-adic realm. Robba also conjectured a $p$-adic interpolation lemma for the case where the set of points of interpolation is what he calls a perturbation of a product set, a situation which includes both the case of a well distributed set and the case of a cartesian product. In this paper, we present an algebraic proof of Robba's conjecture together with a generalization of it over the complex numbers.