Separable Banach spaces which are absolute Lipschitz retracts

Question

A subset $F$ of a metric space $M$ is called a Lipschitz retract of $M$ if there is a Lipschitz map from $M$ onto $F$ which coincides with the identity on $F$. A metric space which is a Lipschitz retract of any metric space containing it is called an absolute Lipschitz retract.

The typical example of separable Banach space which is an absolute Lipschitz retract is $c_0$. More generally, $C_u(M)$ (the space of real-valued, uniformly continuous bounded functions on the metric space $M$ with the sup norm) is an absolute Lipschitz retract (see e.g. Benyamini and Lindenstrauss' book). If $M$ is compact, then $C_u(M)=C(M)$ is separable, but if it is not compact it is not separable since it contains $\ell_\infty$.

Question 1. What other separable Banach spaces which are absolute Lipschitz retracts do we have, except for $C(K)$ spaces with $K$ compact metric?

This question can be considered quite "wiki". My second related question is more specific.

Question 2. Suppose that $X$ is a Banach space which is an absolute Lipschiz retract. Does $X$ contain a copy of $c_0$?

Answer 1

This does not answer the original questions, just something from the comments: if a Banach space is an absolute Lipschitz retract then it is an $\mathcal{L}_\infty$-space.

Let $X$ be a Banach space which is an absolute Lipschitz retract. Let $Y \subset Z$ be Banach spaces and $t : Y \to X$ a bounded linear map. Since $X$ is an absolute Lipschitz retract, there is a Lipschitz extension $\tau : Z \to X$ of $t$. By Theorem 7.2 in the Benyamini-Lindenstrauss book, there is a bounded linear map $T : Z \to X^{**}$ that coincides with $\tau$ on $Y$, so then $T$ is in fact a linear extension of $t$.

Now, by a result of Lindenstrauss (Theorem 2.1 in Extension of compact operators. Mem. Amer. Math. Soc. No. 48 1964), this implies that $X^{**}$ is an injective space. But then both $X^{**}$ and $X$ are $\mathcal{L}_\infty$-spaces (Theorem F.2(v) in Benyamini-Lindenstrauss).

Note: this is an adaptation of the arguments at the beginning of the proof of Proposition 3.5 in Avilés, Antonio; Cabello Sánchez, Félix; Castillo, Jesús M. F.; González, Manuel; Moreno, Yolanda On separably injective Banach spaces. Adv. Math. 234 (2013), 192–216.