Nonseparable C(K)-spaces can be twisted when K is a finite height compact

We show that, given a nonmetrizable compact space K having ω  -derived set empty, there always exist nontrivial exact sequences 0→c0→E→C(K)→00→c0→E→C(K)→0. This partially solves a problem posed in several papers: Is Ext(C(K),c0)≠0Ext(C(K),c0)≠0 for K a nonmetrizable compact set?