Is it obligatory to pay zakat on the initial amount or on the current value of the car, given that purchasing the car led to installments not covered by savings, and that paying zakat would necessitate borrowing?

There is no Zakat on a car that was not purchased with the intention of trade, due to the Prophet's (peace and blessings be upon him) saying: "There is no sadaqah (Zakat) on a Muslim for his slave or his horse." However, if you purchased it for trade, then assess its value when a full Hijri year has passed, and give one-quarter of one-tenth (2.5%) of its value. If you have a debt, subtract it from the money subject to Zakat according to many scholars, but it is more cautious to pay Zakat on the full amount without deducting the debt, to avoid scholarly differences.