In a finite commutative ring , every prime ideal is maximal?

There is no counterexample, because even if the ring has no identity, the quotients by primes must have identity.

Every nonzero finite ring without zero divisors has a multiplicative identity, so the quotient would in fact be a finite domain with identity, and hence a field.

The answer is false. $I$ is prime means $R/I$ is a domain. Which implies $R/I$ is a field which implies that $I$ is maximal.