I asked this question at MSE but I did not received any answer, so I repeat it here at MO:

What is an example of a Hausdorff topological space $X$, not a singleton, such that the ring $C(X)$ of all real (or complex) valued continuous functions on $X$ is a **clean ring**?

A clean ring is a ring in which every element is a sum of a unit and an idempotent.