In the excellent book "Algebraic Geometry 1" of Görtz & Wedhorn, in exercise 3.14, one is asked to show that in the spectrum of a valuation ring with infinitely many primes, the complement of the unique closed point is an open set without a closed point.
It seems to me this is quite not true : to find a counter example, it is sufficient to build a valuation ring with a numerable descending chain of prime ideals, which can be done by using as value group $\mathbb{Z}^{\mathbb{N}}$ with the reverse lexicographical order.
Did I miss something ?