The problem that I posted was a much generalized form of what I had in my mind. All I want to know the literature of Hahn-Banach type extension of Lipschitz map. I know only about the result by Kirszbrzun and very few of its development. The main problem is: Let $X$ be a real Banach sp, $Y$ a subspace of it and $f$ a $1$-Lip map from $X$ to $\mathbb{R}$. Is it possible to get an extn of $f$ from $X$ to $\mathbb{R}$ with Lip constant $1$? The case when $X$ is a metrizable tvs  and $Y$ a finite dim. subsp, the result follows from MR0737400(86a:46018). But can we say anything about a infinite dimensional $Y$? Let us assume $X=C(K)$, $K$ is cpt, $T_2$(can assume metrizable also) and $Y=\{f\in X:f|_D=0\}$ (ie an M-ideal) of $X$. Now can a real valued $1$-Lip map from $Y$ necessarily has a similar extn ? Can it be unique ?