Is it possible for a car to move in a circular path in such a way that it has a tangential acceleration but no centripetal acceleration?

No, it is not possible

Explanation:

The centripetal acceleration of the car is the acceleration of the car in uniform circular motion.

It is this centripetal acceleration that keep the body in circular motion.

The centripetal acceleration is given as

a = v²/r

it is not possible for an object to move in a circular path without centripetal acceleration.

Since, the car is moving in a path that is circular; the car must have a centripetal acceleration.

When a car is moving in circular path, it's velocity is changing direction and for this to be possible to must have an acceleration which is non-zero.

Thus it is impossible for a car moving in a circular path to have tangential acceleration but no centripetal acceleration.