Lipschitz subspaces of C(K)

Let K   be an uncountable metric compact space. It is well known that C(K)C(K) is isometrically universal for the separable Banach spaces, but the continuous functions that compose the isometric image of finite dimensional spaces are typically far from being Lipschitz. We prove that the possibility of embedding Euclidean spaces Rn↪C(K)Rn↪C(K) in such a way that the image in C(K)C(K) is made of Lipschitz functions is tightly related to the dimension (topological or Hausdorff) of K.