To apply for a driving license in Bahrain, you must submit an application to the General Directorate of Traffic, which falls under the Ministry of Interior, and can be found in Isa Town in the north of the country.
Their website advises that the process should take roughly 3 working days, and notes that there isn't a fee for applying for a license.
It is important that you include all the right documentation, of course. And whether you're buying or simply renting a vehicle whilst you're in Bahrain, you'll need an application letter, a copy of personal ID, and a description of what you're doing with the transportation.
Bahrain allows drivers that hold a license in many countries to apply for a Bahrain driving license without having to take a test. There is, however, a small number of countries whose citizens might need to take a test during the application procedure, so it's important to check this in advance when thinking about hiring or buying a car there.