Is it permissible to mortgage a family home to a non-Islamic bank to purchase another home in the city of residence, knowing that Islamic banks are unavailable and the head of the family needs to pay off large debts?

If borrowing from the bank involves mortgaging the house in exchange for an increase (interest), then it is a usurious loan, forbidden by consensus. If the loan is without an increase, but the bank benefits from the house free of charge, this is also forbidden because it is a loan that draws benefit. However, if the loan is repaid without an increase, and the bank does not benefit from the house, or benefits from it for a fair rent, then this is permissible. It is not permissible to enter into a forbidden loan, even out of necessity, due to the severe warning associated with it.