Context. Studing a problem in machine-learning, I'm led to consider the following problem in RMT...

Fix $\delta \in (0, 1)$ once and for all.

Q: Is it possible to find universal constants $c_1 > 0$ and $c_2 > 0$ (depending only on $\delta$) such that the following phenomenon holds ?

Phenomenon. Let $m$ and $n$ be positive integers with $m \le \delta n$ and $n$ large, and let $k \ge c_1 m$. Let $X$ an $m$-by-$n$ random real matrix with iid $N(0,1)$ entries. Finally, denote by $c_2\mathbb B^k$ the ball of radius $c_2$ in $\mathbb R^k$.

With high probability, every $k$-by-$n$ submatrix $Z$ of $X$ verifies $$ c_2\mathbb B^k \subseteq Z\mathbb B^{n}:= \{Zv \mid v \in \mathbb B^n\}. $$

Also, how large can this probability be as a function of $n$ and $\delta$ ?