Let $R = \mathbb Q_p (\mu_p)$ and $k = \mathbb F_p$ it's residue field. Consider $A = R[x]/(x^p-1)$. Then, I believe $A = \prod_{i=0}^{p-1}R[x]/(x-\zeta_p^i)$ so $\operatorname{Spec}A$ has $p$ connected components.
On the other hand, after reducing mod $p$, $A_k = k[x]/(x-1)^p$ so this has only one connected component. This seems to contradict what Conrad says here: https://mathoverflow.net/a/16136/58001, in the paragraph beginning "OK, now assume R is a complete...".
In particular, he claims that for a complete local ring, the connected components of a finite scheme over $R$ and over the residue field are in bijection.
What am I doing wrong?