Is it possible to have a zero knowledge proof with a P Prover?
If the prover has the same power as the verifier, then the verifier can simulate the entire interaction himself, which makes the interactive proof unnecessary. For example, if we have an interactive proof with a probabilistic polynomial verifier and a polynomial prover for some language $L$, then $L\in BPP$. When dealing with interactive proof systems with a polynomial verifier it suffices to consider PSPACE provers, since a PSPACE prover can find and send the messages that maximize the verifier's acceptance probability.
Note that the above refers to interactive proof systems in general, there's nothing specific to zero knowledge (zero knowledge simply adds the condition that for yes instances, the entire communication of the protocol can be generated by a simulator who only has access to the input).