How can a girl who is convinced of the obligation of niqab and strongly desires to wear it deal with her mother's strong refusal and her prohibition of discussing this matter, especially since the mother sees it as extremism, particularly after the girl's decision to withdraw from her studies due to [gender] mixing?

We have previously discussed obedience to parents regarding the removal of the niqab and co-education in fatwas 47072 and 40303. It is recommended to treat your mother gently, honor her, be kind to her, and endure her ill-treatment to gain her affection and avert her harm, as Allah says: "And not equal are the good deed and the bad. Repel [evil] with that which is better, and immediately your enemy will become as a devoted friend." You can seek help from someone whose words you hope your mother will accept in convincing her. Do not forget supplication (du'a), for it is the best helper in achieving one's aim, as Allah says: "And when My servants ask you, [O Muhammad], concerning Me - indeed I am near. I respond to the invocation of the supplicant when he calls upon Me. So let them respond to Me [by obedience] and believe in Me that they may be [rightly] guided." The more the etiquettes of supplication are observed, the more likely the response will be. You can review the etiquettes of supplication in fatwa 97273.