The Knaster-Kuratowski-Mazurkiewicz Lemma is the continuous analogue of Sperner's lemma. I wonder if the following, more general version is true.

Let S be the standard simplex spanned by the standard orthonormal basis for $\mathbb R^{n+1}$, so S equals the convex hull of $(e_i:i\in [n+1])$. Assume we have n+1 closed subsets $C_1, \ldots, C_{n+1}$ with the property that for every subset $I$ of [n+1] the following holds: the convex hull of $(e_i:i\in I)$ is disjoint from $\cup_{i\notin I} C_i$. Is it true that either there are t $C_i$'s that intersect OR there is a k-dimensional "subspace" of S that is disjoint from all the $C_i$'s, if n is large enough (compared to t and k)?

By k-dim subspace of S I mean a linear subspace (passing through the origin) whose intersection with S is k-dimensional. Any better formulations of the problem and retags are welcome.

Note that for k=0 we get back the KKM lemma. I do not know the answer already for k=1. In case it is false, is it possible to replace the affine subspace by something else?

Edit: As Ilya pointed out in the Edit part of his answer, we cannot hope for a k-dim subspace. Any other reasonable "big" manifold we can hope for?