Here is one way you can prove that a space $X$ is not an absolute extensor for compacta: Find sequences $(E_n)$ and $(F_n)$ of finite dimensional subspaces of $X$ and a constant $C$ so that for every $n$ there is a linear isomorphism $T_n$ from $E_n$ onto $F_n$ s.t. $\|T_n\|=1$, $\|T_n^{-1}\|\le C$, each $F_n$ is the range of a projection of norm at most $C$, but the $E_n$ are not uniformly complemented. If the restriction of $T_n $ to the unit ball $B_{E_n}$ of $E_n$ has a $D$ Lipschitz extension from $B_X$ into $X$, then it has a $CD$-Lipschitz extension into $F_n$, and the positively homogenous extension of this mapping is a $3CD$-Lipschitz extension of $T_n$ to a mapping from $X$ into $F_n$. By Lindenstrauss' 1964 paper on non linear projections (see the book of Benyamini-Lindenstrauss), there is then a linear extension $S_n:X \to F_n$ with $\|S_n\|\le 3DC$ so that $T_n^{-1}S_n$ is a projection from $X$ onto $E_n$ having norm at most $3DC^2$. So if such an X is an absolute extensor for compacta, there is no uniform Lipschitz bound on extensions of non expansive linear mappings from finite dimensional subspaces.

It remains to show that if a space $X$ is an absolute extensor for compacta, then there is a uniform bound on extensions of non expansive linear mappings defined on finite dimensional subspaces. Suppose there is no such uniformity for $X$. Notice that then if $Y$ is a finite codimensional subspace of $X$, then there also is no uniformity for extensions of non expansive linear mappings defined on unit balls of finite dimensional subspaces of $Y$ into $X$. Now use the Mazur technique for constructing basic sequences to build a finite dimensional Schauder decomposition $(E_n)$ for some subspace of $X$ and non expansive linear mappings $f_n: E_n \to X$ s.t. any extension of $f_n$ to $X$ has Lipschitz constant at least $n $. Let $K = \cup_n n^{-1} B_{E_n}$ and define $F:K\to X$ by $f(x) = f_n(x)$ if $x\in n^{-1} E_n$. Then $K$ is compact and $f$ is Lipschitz and $f$ has no Lipschitz extension to $X$.

There are many spaces that contain sequences $(E_n)$ and $(F_n)$ of finite dimensional subspaces for which there is a constant $C$ so that for every $n$ there is a linear isomorphism $T_n$ from $E_n$ onto $F_n$ s.t. $\|T_n\|=1$, $\|T_n^{-1}\|\le C$, each $F_n$ is the range of a projection of norm at most $C$, but the $E_n$ are not uniformly complemented.
If $X$ is $ \ell_p$ or $L_p$, $1\le p \not=2 <\infty$, then $E_n$ can be taken to be uniformly isomorphic to $\ell_p^n$. If $X$ is super reflexive (or even just has non trivial type) but does not have type $p$ for some $p<2$, then $E_n$ can be taken to be uniformly isomorphic to $\ell_2^n$. These results are fairly deep, BTW, but well known to researchers in Banach space theory.