This question was featured on a qualifying exam at my university:
What's an example of a commutative local ring $R$ of characteristic zero, with a non-maximal prime ideal $P$ such that the characteristic of $R/P$ is not zero?
Our favorite example of a local ring, $\mathbb{Z}_{(p)}$, won't work because it's a PID (a DVR in fact) and won't have any non-maximal prime ideas. I think that the ring of power series $\mathbb{Z}_{(2)}[\![x]\!]$ might be an example, but I haven't worked out the details yet.
Best Answer
An example will be the ring $\mathbb{Z}[x]$ localized at $(x,2)$, so $\mathbb{Z}[x]_{(x,2)}$. An important fact here that makes this a reasonable example to come up is that the localization of a ring $R$ at a prime ideal $P$ will be a local ring $R_P$, the max ideal being $P_P$, and furthermore the prime ideals of $R_P$ will all be of the form $Q_P$ for some prime ideal $Q$ of $R$ that is contained in $P$. So for our particular example, we're looking at the chain of prime ideals $(0) \hookrightarrow (2) \hookrightarrow (2,x) \hookrightarrow \mathbb{Z}[x]$. The ideal $(2)_{(2,x)}$ will be prime in $\mathbb{Z}[x]_{(x,2)}$, and since $\mathbb{Z}[x]_{(x,2)}$ is still unital, the quotient of $\mathbb{Z}[x]_{(x,2)}$ by $(2)_{(2,x)}$ will have characteristic $2$.
The ring $\mathbb{Z}_{(2)}[\![x]\!]$ mentioned in the question is an example too, for nearly the same reason: the ideal $(x,2)$ is maximal and $(2)$ is prime.