Let $p$ be a prime integer and $R$ be a finite local ring. Assume that $p||R^\times|$. Then by Cauchy's Theorem, there always exists a primitive $p$ root of unity in $R^\times$. Here $R^\times$ is a unit group of $R$.
Is it possible to find a finite local ring $R$ which is not a finite field and a prime integer $p>2$ dividing $|R^\times|$ such that there exists a primitive $p$ root of unity $u \in R^\times $ for which $u - 1$ is still a unit in $R$?