Is it permissible to work in banks under the aforementioned circumstances, given that the high salary and short working hours help in supporting the family and continuing one's studies?

If the bank's transactions are based on Islamic Sharia rulings and are subject to Sharia supervision, then there is no harm in working there. However, if its transactions involve usury (riba), then it is not permissible to work there, because it is considered cooperating in sin and aggression. The Messenger of Allah (peace be upon him) "cursed the one who consumes usury, the one who pays it, and its two witnesses, and he said: 'They are all equal (in sin).'" In this case, repentance and leaving the job are obligatory. Lawful earnings are better than unlawful earnings. Your situation is not considered a necessity that permits prohibited work. We advise you to supplicate and not despair.