Let $A$ be an integral domain and $P$ be a prime ideal in $A$. We denote $B=A/P$ and also suppose $P$ is a reflexive $A$-module. Then is the following natural map $$Hom_A(P,A)\otimes_A B\to Hom_A(P,B)$$ surjective?

Any comments or suggestions regarding the question will be helpful. Thanks in advance!!