Yes! See my recent preprint Alternating quotients of free groups.
I expect that what you want was is well known, but I too couldn't find it in the literature. In fact, I prove the much stronger result that free groups are something like 'locally extended fully residually alternating'. Specifically:
Let $F$ be any free group of rank at least two, let $H$ be a finitely generated subgroup of infinite index in $F$ and let
$\{g_1,\ldots,g_n\}$be a finite subset of $F\smallsetminus H$. Then there is a surjection $f$ from $F$ to a finite alternating group such that $f(g_i)$ is not in $f(H)$ for any $i$.
