Is there a specific problem that is in both NP and co-NP but not in P?
An answer to your question would be a problem $p$ which is in $\mathop{NP}\cap\mathop{coNP}$, and thus in $\mathop{NP}$, but not in $P$. Existence of such a problem would easily imply $P\neq\mathop{NP}$, which is a seriously open problem to this date. That is to say, no one can answer your question yet.