Lines of full rank matrices in large subspaces

Let $n$ and $p$ be non-negative integers with $n\ge p$, and $S$ be a linear subspace of the space of all $n$ by $p$ matrices with entries in a field $\mathbb{K}$. A classical theorem of Flanders states that $S$ contains a matrix with rank $p$ whenever $\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}S<n$. In this article, we prove the following related result: if $$\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}$ S<n-1$,then,foranynon-zero$n$by$p$matrix$N$withranklessthan$p$, there exists a line that is directed by $N$, has a common point with $S$ and contains only rank $p$ matrices.