An example of prime ideal $P$ in an integral domain such that $\bigcap_{n=1}^{\infty}P^n$ is not prime

Let $k$ be any field. Set $$R = \bigcup_{n=1}^{\infty} k\left[x,\ y,\ x^{1/n} y^{1/n} \right].$$ Each one of the terms in the union is a domain, so the rising union is also a domain. Let $P$ be the ideal $\langle x, y, x y, x^{1/2} y^{1/2}, x^{1/3} y^{1/3}, x^{1/4} y^{1/4}, \cdots \rangle$. Then $R/P = k$, so $P$ is prime (and even maximal).

We have $P^k = \langle x^k , y^k, x y, x^{1/2} y^{1/2}, x^{1/3} y^{1/3}, x^{1/4} y^{1/4}, \cdots \rangle$ and $\bigcap_{k=1}^{\infty} P^k = \langle x y, x^{1/2} y^{1/2}, x^{1/3} y^{1/3}, x^{1/4} y^{1/4}, \cdots \rangle$. So $R/\bigcap_{k=1}^{\infty} P^k = k[x,y]/(xy)$ which is not a domain, and we see that $\bigcap_{k=1}^{\infty} P^k$ is not prime.