  In this paper, we prove that the Cauchy problem for a generalized
Camassa-Holm equation with higher-order nonlinearity is ill-posed in the
critical Besov space $B^1_{\infty,1}(\R)$. It is shown in (J. Differ. Equ.,
327:127-144,2022) that the Camassa-Holm equation is ill-posed in
$B^1_{\infty,1}(\R)$, here we turn our attention to a higher-order nonlinear
generalization of Camassa-Holm equation proposed by Hakkaev and Kirchev (Commun
Partial Differ Equ 30:761-781,2005). With newly constructed initial data, we
get the norm inflation in the critical space $B^1_{\infty,1}(\R)$ which leads
to ill-posedness.
