  For a metric space $(K,d)$ the Banach space $\Lip(K)$ consists of all
scalar-valued bounded Lipschitz functions on $K$ with the norm
$\|f\|_{L}=\max(\|f\|_{\infty},L(f))$, where $L(f)$ is the Lipschitz constant
of $f$. The closed subspace $\lip(K)$ of $\Lip(K)$ contains all elements of
$\Lip(K)$ satisfying the $\lip$-condition $\lim_{0<d(x,y)\to
0}|f(x)-f(y)|/d(x,y)=0$. For $K=([0,1],| {\cdot} |^{\alpha})$, $0<\alpha<1$, we
prove that $\lip(K)$ is a proper $M$-ideal in a certain subspace of $\Lip(K)$
containing a copy of $\ell^{\infty}$.
