If $(X,\tau)$ is a topological space we say that an open cover $\mathcal{U}$ is a *clopen partition cover* if it consists of disjoint clopen sets. Trivially, every clopen partition cover is locally finite.

Is there a paracompact space $(X,\tau)$ such that

- $(X,\tau)$ is zero-dimensional, that is for $x,y\in X$ there is $U\subseteq X$ clopen such that $x\in U, y\notin U$, and
- there is an open cover $\mathcal{U}$ of $X$ such that $\mathcal{U}$ does not have a refinement that is a clopen partition cover

?