Is it possible for a female to become a Super Saiyan, or is it restricted to males? Could a fully Saiyan female (e.g. if Goku was born a girl) become a Super Saiyan? I know that Pan is one quarter Saiyan; perhaps that isn't enough to transform. Could she still change if she gathered enough Ki during battle?
No female Super Saiyans are ever seen in the manga/anime. However, the Dragon Ball GT Perfect Files implies that Pan has the potential to transform and might have become a type of Super Saiyan never seen before. Female Super Saiyans have since appeared in video games, including Dragon Ball Online and Dragon Ball Heroes. "
So to my understanding it is very possible for a female to become super saiyan in anime/manga, and definitely possible in video games. Yes, your female Goku would be able to achieve super saiyan state because if Pan is able to become a super saiyan, whether or not it is a new form it would still be considered super saiyan. (:
While the extent is not yet know, Dragon Ball Super has introduced the first female Super Saiyan, Caulifla. Although she is from Universe 6 (where Saiyans were good, not evil), she is still a Saiyan
It took Cabba telling her about the Super Saiyan form as well as about being able to achieve greater heights in her strength to make her agree, showing her desire to become stronger.
She also got angry at Cabba for his attempts at provoking her, even to have her obtain the Super Saiyan form. After obtaining the Super Saiyan transformation, Caulifla seemed grateful for Cabba teaching her.