Examples for exact tests which are not permutation tests

Wikipedia states (without citation):

Sometimes permutation test is used as a synonym for exact test, but although all permutation tests are exact tests, not all exact tests are permutation tests.

What are examples of exact tests which are not permutation tests? I assume that Fisher's exact test is also considered to be a permutation test since it looks to my understanding at all possible permutations of contingency tables (see also here). Is this correct?