Is it permissible to sell a car in installments after purchasing it in cash, designating it in the showroom, writing a سند (document/deed) with the amount and chassis number, then selling it to the buyer and documenting that with a كمبيالة (promissory note)?

It is not permissible to sell a purchased item before taking possession of it, whether it is movable or immovable. This is based on the Prophet's (peace be upon him) saying: "Whoever buys food should not sell it until he has taken full possession of it." Possession is achieved by moving the purchased item from its place to another place that is not exclusive to the seller, or to him with his permission. So, if you wish to sell a car you have purchased, it must be moved from its place, even to another place within the same showroom if the seller permits it. This is the view of the Shafi'is and Hanafis, and one of the two narrations from Ahmad.