  In this paper, we are concerned with bilinear estimates related to the
two-dimensional stationary Navier--Stokes equation. By establishing concrete
counter examples, we prove the bilinear estimates fail for almost all scaling
critical Besov spaces. Our result may be closely related to an open problem
whether the two-dimensional stationary Navier--Stokes equation on the whole
plane $\mathbb{R}^2$ is well-posed or ill-posed in scaling critical Besov
spaces.
