What is the ruling on a borrower leaving the country before fully repaying the value of a car purchased through an Islamic bank, while committing to paying the installments from abroad, and is it permissible for him to do so until the bank sells the car at auction? What is the solution if he cannot repay the full amount?

You must adhere to the contract signed with the Islamic bank and not violate it, and not harm others (the guarantors). The best solution is direct understanding with the bank to settle the debt, either by selling the car or bringing a guarantor for the remaining amount, or continuing with monthly payments. If the contract stipulates not leaving the country before settling the installments, then it is not permissible to leave permanently or temporarily with the intention of not returning. You must settle the situation with the bank before leaving the country or be patient until the debt is paid.