I must admit that I know nothing about p-adic geometry, so the following question may be completely trivial.

Let $V\subset K^n$ be an affine algebraic variety. Let $D$ be a polydisk, and $F$ be an analytic map $f:D\to K^m$. Suppose that for every smooth point in $V\cap D$ the rank of $df|_V$ is not exceeding $k$. My goal is to estimate the dimension of fibers $V\cap f^{-1}(f(z))$. It's trivial that at smooth points the dimension of fibers is at least $n-k$. Is it also true for singular points?

If $K=\mathbb{C}$, this is follow from the semicontinuity of the fiber dimension (proved, for example, in Chirka "Complex analytic sets"). Is it also true for $K=\mathbb{C}_p$?

I am looking for the reference (preferably), proof, or some ideas.