  For every functional functor $F:Comp\to Comp$ in the category $Comp$ of
compact Hausdorff spaces we define the notions of $F$-Dugundji and $F$-Milutin
spaces, generalizing the classical notions of a Dugundji and Milutin spaces. We
prove that the class of $F$-Dugundji spaces coincides with the class of
absolute $F$-valued retracts. Next, we show that for a monomorphic continuous
functor $F:Comp\to Comp$ admitting tensor products each Dugundji compact is an
absolute $F$-valued retract if and only if the doubleton $\{0,1\}$ is an
absolute $F$-valued retract if and only if some points $a\in F(\{0\})\subset
F(\{0,1\})$ and $b\in F(\{1\})\subset F(\{0,1\})$ can be linked by a continuous
path in $F(\{0,1\})$. We prove that for the functor $Lip_k$ of $k$-Lipschitz
functionals with $k<2$, each absolute $Lip_k$-valued retract is openly
generated. On the other hand the one-point compactification of any uncountable
discrete space is not openly generated but is an absolute $Lip_3$-valued
retract. More generally, each hereditarily paracompact scattered compact space
$X$ of finite scattered height $n$ is an absolute $Lip_k$-valued retract for
$k=2^{n+2}-1$.
