Is there a Johnson-Lindenstrauss Lemma for large finite fields?

The Johnson-Lindenstrauss lemma shows that points in a high-dimensional space over the real numbers (I think also works for natural numbers) can be mapped into a lower dimensional space, such that pairwise distances are approximately preserved.

Question: Is there an equivalent lemma for vector spaces over a large finite field of prime order? If not, is there some intuitive reason for why it does or cannot exist?

A related question was already asked, but the answers seemed to be exclusively concerned with the vectors over $ \mathbb{F}_2$ .