If $pi: Y to X$ is an unramified double cover of a smooth curve of genus $g$, then the Prym variety $P_pi$ is a principally polarized abelian variety of dimension $g-1$. When $X$ is defined over an algebraically closed field $k$ of characteristic $p$, it is not known in general which $p$-ranks can occur for $P_pi$ under restrictions on the $p$-rank of $X$. In this paper, when $X$ is a non-hyperelliptic curve of genus $g=3$, we analyze the relationship between the Hasse-Witt matrices of $X$ and $P_pi$. As an application, when $p equiv 5 bmod 6$, we prove that there exists a curve $X$ of genus $3$ and $p$-rank $f=3$ having an unramified double cover $pi:Y to X$ for which $P_pi$ has $p$-rank $0$ (and is thus supersingular); for $3 leq p leq 19$, we verify the same for each $0 leq f leq 3$. Using theoretical results about $p$-rank stratifications of moduli spaces, we prove, for small $p$ and arbitrary $g geq 3$, that there exists an unramified double cover $pi: Y to X$ such that both $X$ and $P_pi$ have small $p$-rank.
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