My new question is again in the context of Hausdorff 0-dimensional spaces.
We say that S subspace of a space X is a 2-embedding if for every continuous function with domain S and codomain 2(the discrete space of two points) there is a continuous extension for the function defined in the total space and codomain 2, and if we put $\mathbb{N}$ instead 2 we obtain the definition of $\mathbb{N}$-embedding.
Then the problem is this: Let S a 2 embedding of a space X. Then S is a $\mathbb{N}$-embedding if and only if for every F such that is disjoint with S and is the intersection of a countable family of clopen subsets of X, there exist a clopen subset T of X such that $F\subseteq T$ and $S\subseteq X/T$.
Thank you for the further answers.
We can use the same reformulation as in Embeddings and discrete spaces. A continuous function $f: X \to \mathbb{N}$ is actually the same thing as clopen decomposition of $X$ indexed by $\mathbb{N}$. If you have $f$ you can take $\{C_i = f^{-1}[\{i\}]: i ∈ \mathbb{N}\}$. If you have $\{C_i: i ∈ \mathbb{N}\}$ you can define $f$ so that $f$ takes value $i$ on $C_i$.
For one implication, we start with a decomposition of $S$ into countable clopen subspaces $\{C_i: i < ω\}$. And we want to extend this decomposition to clopen decomposition of whole $X$. Because $S$ is $2$-embedded we can find for any $i$ a clopen subset of $X$ $C'_i$ such that $C'_i ∩ S = C_i$. Now we have decomposition of $X$ into sets $\{D_i = C'_i ∩ \bigcap_{j < i} (X \setminus C'_j): i < ω\}$ and $D_ω = \bigcap_{j < ω} (X \setminus C'_j)$. But the last set $D_ω$ is not necessarily clopen which is our last obstacle. However it is disjoint with $S$ and is a countable intersection of clopens so we can use the second assumption and find that clopen $T$ which separates $S$ from $D_ω$. Now the extended decomposition of $X$ which we are looking for is made of $D_i ∩ T$ for $i < ω$ and $X \setminus T$. It is actually made of one more element that the original decomposition of $S$ but it doesn't matter since we can add this extra component $X \setminus T$ to any other component.
I don't know how to prove the other implication. We have $\{C_i: i < ω\}$ clopen subsets in $X$ such that $S ∩ C = ∅$ where $C = \bigcap_{i < ω} C_i$ and we want to conclude that $S$ can be clopenly separated from $C$ under assumption that $S$ is $\mathbb{N}$-embedded into $X$. Let's take $\{\bigcap_{j < i} C_j ∩ (X \setminus C_i) ∩ S: i < ω\}$ which is by assumption clopen decomposition of $S$. By the assumption this can be extended to clopen decomposition of $X$. If $C$ intersected only finite number of components of this decomposition (which it would if $X$ was countably compact) than one could take their union intersected with enought sets $C_i$.
Also note that for countably compact $C$ can be clopenly separated from any countable intersection of clopen sets, so countably compact $2$-embedded subspace is $\mathbb{N}$-embedded.