In Michael Coornaert's book 'Topological Dimension and Dynamical Systems' he mentions the existence of a set $X$ in the plane that is totally separated, but not zero-dimensional. He says it is due to Sierpinski, but the paper he cites is in French. The example doesn't seem to be in Steen & Seebach; is anyone familiar?
Here, 'totally separated' means that the quasicomponents of $X$ are points. Equivalently, for any $x, y \in X$ there is a separation of $X$ into disjoint open sets $U, V$ such that $x \in U$ and $y \in V$.
The Sierpinski paper is here: https://eudml.org/doc/212954
A zero-dimensional space in this context can equivalently mean a space with a clopen basis, a space all of whose points have a clopen basis, or a space with arbitrarily fine covers by disjoint open sets.
EDIT: David Hartley below has described Sierpinski's space satisfying the desired requirements. I've attached a (bad) drawing.