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 $c_0$?