What is an example of a 0-dimensional locally compact Hausdorff space $X$ for which the Stone-Čech compactification $\beta(X)$ is *not* 0-dimensional?

It is known that if $X$ is a 0-dimensional locally compact Hausdorff space which is also paracompact, then $\beta(X)$ is 0-dimensional. (Engelking 1989, Th. 6.2.9). I would expect a counterexample in the non-paracompact case.

Another way of asking the question is to look at a Boolean ring L (without assuming a unit). If X is the Stone space of L then X is a 0-dimensional locally compact Hausdorff space and L is isomorphic with the ring of compact open sets in X. The Boolean algebra $\mathrm{Cl}(X)$ of closed-open subsets is clearly a Boolean algebra extension of L, and the Stone space of $\mathrm{Cl}(X)$ is a certain compactification of $X$. I believe it is easy to see that this compactification of $X$ is just $\beta(X)$ if and only if $\beta(X)$ is 0-dimensional.

Thus the question can be expressed in the algebraic side of the duality. Find conditions on the Boolean ring $L$ so that $\beta(X)$ is 0-dimensional, where $X$ is the Stone space of $L$.

I think this may be true if, for example, $L$ is a Boolean $\sigma$-ring. My original question asks for an example showing this is not true without some additional conditions on $L$.