Lipschitz spaces and M-ideals

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