All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
The prohibition of hitting the face is not restricted to fighting; rather, it is forbidden in all cases.
Abu Hurayrah, may Allah be pleased with him, narrated that the Prophet said, “If one of you were to hit, he should avoid the face...” [Musnad Ahmad]
As-San'aani said in Subul As-Salaam:
“This is evidence on the prohibition of hitting the face, and it means that one should avoid hitting or slapping the face, even if this is regarding one of the Islamic Hudood (plural of Hadd, i.e. the corporal punishment determined by the Islamic Law), and even in jihaad, because the face encompasses all beauties; its parts are sensitive and include most of the senses of perception. Hitting the face may cause defects to them or impair them, and it may cause disfigurement. Disfigurement in the face is grave because it is visible and cannot be concealed. When the face is hit, it is rarely spared from deformity. This forbiddance (to hit the face) is general regarding any beating or slapping regardless of whether it is in order to discipline or else.”
Since this act is forbidden in jihaad, then with greater reason its prohibition becomes applicable to cases other than jihaad, among which is self-defense. The mention of jihaad proves the prohibition of this act even when fighting against the Kaafir (non-Muslim), and it shows that this is not restricted to fighting the Muslim. Also, it should be noted that repelling the attacker should be done with the easiest, and then the less easy method, as stated by the scholars.
With regard to the word 'fight' that is mentioned in a relevant hadeeth, Al-‘Iraaqi said in Tarh At-Tathreeb:
“If fighting occurs between two parties, even when repelling an attacker and the like, he should avoid hitting his face; even more so if the other party is not hitting him. In this latter case, it is with greater reason that it becomes more applicable to avoid hitting his face, because the one who is defending himself may be forced to hit the face; however, despite this fact, it is prohibited in the Sharia to hit the face, so it becomes applicable to the one who is not fought by someone else, with greater reason, that he is ordered to avoid hitting his face.”
All this is evidence that hitting the face is forbidden regardless of whether there is a sound justification for fighting.
Some scholars exempted jihaad in particular and permitted hitting the face for a necessity. Mulla ‘Ali Qaari said in Mirqaat Al-Mafaateeh, “There is no doubt that this is not meant in all cases, because we can authoritatively assert that in case of war with the non-Muslims, if one is in a situation where he has to hit the face of the one who is fighting with him, or when he is face to face with him when he is attacked, he does not withhold himself from hitting him. If he withheld himself, he could be killed by the attacker.”
Some scholars exempted other parts of the body from hitting, such as the private parts, as ‘Ali, may Allah be pleased with him, said, “Avoid hitting the face and the private parts.” [Ibn Abi Shaybah]
Some scholars also forbade hitting the head, while others permitted it, as ‘Umar, may Allah be pleased with him, hit Sabeegh on his head; this story was reported by Ad-Daarimi in his Musnad.
There is nothing wrong with learning martial arts (self-defense techniques) provided that this does not involve what is Islamically prohibited. Please refer to fatwas 308272 and 82646.
Allah knows best.