I am looking for an example of a noetherian local ring $A$ such that its formal fibers are not complete intersection rings in the sense of https://stacks.math.columbia.edu/tag/09PY (i.e., there is a prime ideal $\mathfrak{p}\subset A$ such that the ring $k(\mathfrak{p}) \otimes_A \widehat{A}$ is not a complete intersection ring).

I assume that there should be a classical reference for such example, but I cannot find it online.

P.S. I would appreciate a lot an example that is "easy to digest" (if such exists at all).

P.S.2 Clearly, any such ring cannot be an excellent ring. However, a classical example of a non-excellent DVR does have lci formal fibers, so this example does not work.