This transaction is called “muraababah initiated by the purchaser”, and there is nothing wrong with it, provided that the bank owns the car and takes possession of it before selling it to the customer, and that the contract is free of any stipulation of penalties in the event of late payment, and that the purchaser does not sign a contract obliging him to buy from the bank. This has been discussed in the answer to question no. 129312 and 140603
With regard to the down payment made by the customer before drawing up his contract with the bank and even before the bank takes possession of the car, if this down payment will not be returned in the event of his changing his mind, then this comes under the heading of consuming wealth unlawfully, because the down payment – according to those who regard it as valid, namely the Hanbalis – is only binding by means of the contract and not by mere promises. This has been discussed in the answer referred to above.
But if what is meant by the down payment is confirming serious intent to buy and it will not be withheld from the customer in the event of his changing his mind about buying, apart from an amount of money to compensate for actual harm, then this is permissible according to a number of scholars.
It says in al-Ma‘aayeer ash-Shar‘iyyah (p. 134): It is permissible for the bank to require the one who intends to rent something to pay a certain amount to the bank which it then keeps as a guarantee of commitment on the customer’s part to carry out his promise of renting and other related obligations, provided that nothing is to be deducted from it except an amount that is commensurate with actual harm so that – if the customer changes his mind – he will then be charged the difference between the agreed rent of the item that he was going to rent and the actual rent fees collected by renting it to someone else, or – in the case of selling – he will be charged the difference between the price he had agreed to and the price for which it was sold (to someone else). This down payment to demonstrate commitment is like a trust to be kept with the bank, in which case it is not allowed to dispose of it, or it may be regarded as a trust to be invested with the permission of the customer, on the basis of a legitimate mudaarabah (shared profit) agreement between the customer and the bank. It is also permissible to make an agreement with the customer, when the rental contract is drawn up, to regard this money as part of the instalments.
On p. 147 it says: The basis for the bank requesting a sum of money from the one who wants to rent is the need to confirm his commitment, because the promise of commitment will result in financial repercussions if he changes his mind and the bank will be compelled to ask him to pay for losses or damage, and he may delay such payment. A fatwa concerning this matter (down payment) has been issued by the United Sharee‘ah Council of al-Barakah. This also applies to renting. End quote.
There is nothing wrong with leaving the car registered in the name of the bank until the customer has paid off all instalments. This comes under the heading of collateral which is kept until the price has been paid. But we confirm that the customer becomes the owner of the car as soon as the contract is drawn up.
See also the answer to question no. 69877.
And Allah knows best.