  We show that the incompressible Euler equations on $\mathbb{R}^2$ are not
locally well-posed in the sense of Hadamard in the Besov space
$B^1_{\infty,1}$. Our approach relies on the technique of Lagrangian
deformations of Bourgain and Li. We show that the assumption that the
data-to-solution map is continuous in $B^1_{\infty,1}$ leads to a contradiction
with a well-posedness result in $W^{1,p}$ of Kato and Ponce.
