This question is similar to a previous one: Gauss Elimination with constraints, but it is different.
Given an $n \times n$ matrix $M$ and a number $1 \leq m \leq n-1$, we partition M as a block matrix: $ M=\left[ \begin{array}{cc} A & B \\ C & D \end{array}\right] $ where $A$ is an $m \times m$ matrix and $D$ is an $(n-m) \times (n-m)$ matrix. We then say that $M$ is m-good if both $A$ and $D$ are invertible.
The question: Given any invertible matrix $M \in GL_n(\mathbb{F})$ and a number $1 \leq m \leq n-1$, is it always possible to permute the rows of $M$ to make it m-good?
Note: I only care about the case $\mathbb{F}=\mathbb{Z}_p$, but I asked the question more generally because my feeling is that it doesn't matter what the field is.