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](http://en.wikipedia.org/wiki/Paracompact_space) 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


?



Paracompact zero-dimensional space without clopen partition refinement
I claim that there are such counterexamples. Any Hausdorff space where every open cover can be refined by a partition into clopen sets is called ultraparacompact. In the paper Not every O-dimensional realcompact space is N-compact by Peter Nyikos, he shows that a certain zero-dimensional metrizable space is paracompact but not ultraparacompact and not even realcompact. That being said, there are several characterizations of which spaces are ultraparacompact. In particular, the ultraparacompact spaces are precisely the strongly zero-dimensional paracompact spaces (recall that a space is strongly zero-dimensional if its Stone-Cech compactification is zero-dimensional). I gave more characterizations of ultraparacompactness in [this long answer](https://mathoverflow.net/a/134184/22277).


