TITLE: Between Tietze's and Dugundji's Extension Theorems
QUESTION [12 upvotes]: The celebrated Tietze Extension Theorem asserts that any continuous real-valued function defined on a closed subset of a normal space, can be extended to a continuous function on the whole space. Seen as a property of the target space $\mathbb{R}$, this leads to the important notion of absolute neighborhood retract, or AR(normal), in Dugundji's notation; Tietze Extension Theorem can thus be rephrased saying that   $\mathbb{R}$ is an AR(normal) space. 
If in Tietze theorem we restrict the class of domains from normal to metric spaces, by the Dugundji Extension theorem, at least all locally convex topological vector spaces are suitable  codomains: any continuous LCTVS-valued function on a closed subset of a metric space can be extended to a continuous function on the whole space. 
Of course, this situation in principle allows a wide variety of intermediate situations. The first natural questions, that I would be glad to learn an answer of, are:

Q1. Does Dugundji's theorem hold true for normal spaces, namely, can any continuous LCTVS-valued function on a closed subset of a
  normal topological space be extended to a continuous function on the
  whole space?

I guess the answer is no, but I can't imagine a counterexample. In case of a negative  (or not known) answer:

Q2. Are Banach spaces absolute retract for Hausdorff compact spaces, namely, can any continuous Banach-valued function on a closed subset of a
  Hausdorff compact space be extended to a continuous function on the
  whole space?

edit After Bill Johnson's answer to question 2, and the other useful comments, I would like to focus on the following question, that   should have some good reference in the (wide) literature.

Q3. Let $X$ be a Hausdorff compact topological space, $Y\subset X$ a closed set,  $E$ a Banach space. Does there always exist a bounded
  linear extension operator $C(Y,E)\to C(X,E)$?

REPLY [7 votes]: The following is a rather well-known theorem of Haydon that might be useful for your purpose.

Theorem: The following are equivalent for a compact Hausdorff space $Y$:

($Y$ is a Dugundji space): For every compact $X$ and embedding $e:Y \to X $ there is a bounded linear extension operator $T:C(Y) \to C(X)$ such that $T(1)=1$ and $T(f)\geq0$ whenever $f\geq0$.

($Y$ is $\mathrm{AE}(0)$): For every compact zero-dimensional $Z$, every closed $A \subseteq Z$ and every map $f:A \to Y$ there is an extension $f:Z \to Y$.

$Y$ is the inverse limit of a Haydon system.



See for example Todorcevic´s book "Topics in Topology" for a proof of this theorem (and the definition of Haydon system).
For example any product of compact metrizable spaces is Dugundji and compact topological groups are Dugundji.