Detailed balance for the Fokker-Planck equation
No. To simplify things, we suppose that $\pi$ has a density which we will denote also as $\pi$. The case of general $\pi$ is largely the same but a bit more convoluted as we would have to use distributions.
We suppose for contradiction that $\nabla \times b \neq 0$ and that $\pi$ satisfies $-b(x) \pi + \nabla \pi = 0$. Rearranging this equation we get $$-b(x) = \frac{\nabla \pi}{\pi}. $$ The right-hand side is the gradient of $\log{\pi}$ so it must have zero curl but the left-hand side has non-zero curl by assumption.