  We consider the two-dimensional stationary Navier--Stokes equations on the
whole plane $\mathbb{R}^2$. In the higher-dimensional cases $\mathbb{R}^n$ with
$n \geqslant 3$, the well-posedness and ill-posedness in scaling critical
spaces are well-investigated by numerous papers. However, despite the attention
of many researchers, the corresponding problem in the two-dimensional whole
plane case was a long-standing open problem due to inherent difficulties of
two-dimensional analysis. The aim of this paper is to address this issue and
prove the ill-posedness in the scaling critical Besov spaces based on
$L^p(\mathbb{R}^2)$ for all $1 \leqslant p \leqslant2$ in the sense of the
discontinuity of the solution map and the non-existence of small solutions. To
overcome the difficulty, we propose a new method based on the contradictory
argument that reduces the problem to the analysis of the corresponding
nonstationary Navier--Stokes equations and shows the existence of nonstationary
solutions with strange large time behavior, if we suppose to contrary that the
stationary problem is well-posed.
