Sum-free sets which are closed under multiplicative inverses

Let $A$ be a subset of a finite field $\mathbb{F}$. When $\mathbb{F}$ has prime order, we show that there is an absolute constant $c > 0$ such that, if $A$ is both sum-free and equal to the set of its multiplicative inverses, then $|A| < (0.25 - c)|\mathbb{F}| + o(|\mathbb{F}|)$ as $|\mathbb{F}| \rightarrow \infty$. We contrast this with the result that such sets exist with size at least $0.25|\mathbb{F}| - o(|\mathbb{F}|)$ when $\mathbb{F}$ has characteristic $2$.