C(K) that contains $c_0$ complemented but $K$ does not contain convergent sequence.

Question

Let $K$ be compact Hausdorff topological space, $C(K)$ the Banach space of continuous functions from $K$ in $\mathbb{R}$, endowed with supremum norm. It is known that if $K$ contains convergent sequence then $c_0$ is complemented in $C(K)$. I would appreciate if somebody let me know an example of a compact Hausdorff $K$ without convergent infinite sequences such that $c_0$ is complemented in $C(K)$.

Answer 1

This question has been answered by Tomek Kania at mathoverflow. A example given by he is $\beta \mathbb{N} \times \beta \mathbb{N}$. Thanks to all who try to answer this question here.