Why doesn't a person cease to exist as soon as becoming an arahant?
The mind cannot exist without a body except in the Arupa Brahma realms. 
Arahanths have no desire to die. So becoming enlightened does not imply that they have to immediately die. It's like when you are driving a car, if it runs out of fuel, it's not going to stop immediately. The momentum will take it some distance before stopping.