p-adic interpolation of iterates

Extending work of Bell and of Bell, Ghioca, and Tucker, we prove that for a p-adic analytic self-map f on a closed unit polydisk, if every coefficient of f(x) - x has valuation greater than that of p(1/(p-1)), then the iterates of f can be p-adically interpolated; that is, there exists a function g(x, n) analytic in both x and n such that g(x, n) = f(n)(x) whenever n is an element of Z(>= 0).