On the three space property for $C(K)$-spaces

Assuming $\mathfrak{p}=\mathfrak{c}$, we show that for every Eberlein compact space $L$ of weight $\mathfrak{c}$ there exists a short exact sequence $0\to c_0\to X\to C(L)\to 0$, where the Banach space $X$ is not isomorphic to a $C(K)$-space.