Nonseparable $C(K)$-spaces can be twisted when $K$ is a finite height   compact
  We show that, given a nonmetrizable compact space $K$ having $\omega$-derived set empty, there always exist nontrivial exact sequences $0\to c_0\to E\to C(K)\to 0$. This partially solves a problem posed in several papers: Is $Ext(C(K), c_0)\neq 0$ for $K$ a nonmetrizable compact set? 
Jes\'us M. F. Castillo
math.FA