The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power series, p(z) is a distinguished polynomial (monic, with the coefficients of the non-leading term each in the maximal ideal), and $π$ is a fixed uniformizer.
I would like to know if there is an analogue for this theorem in several variables as one has in the complex case where the distinguished polynomial has the form $a_0(z_1,..,z_n)+za_1(z_1,...,z_2)+..+z^{k-1}a_{k-1}(z_1,...,z_2)+z^k$, each $a_i(z_1,...,z_2)$ being analytic.
I would appreciate any reference you know on this topic.
