In this paper we consider the complexity of computing permanents over fields of characteristic 3. We present a polynomial time algorithm for computing $\Per (A)$ for a matrix $A$ such that $\rank (AA^T - I) \leq 1$. On the other hand, we show that existence of a polynomial-time algorithm for computing $\Per (A)$ for a matrix $A$ such that $\rank (AA^T - I) \geq 2$ implies $\bnp = \br$. As a byproduct we obtain that computing $\Per (A)$ for a matrix $A$ such that $\rank (AA^T - I) \geq 2$ is $Mod_3\bP $-complete.