Examples of an integral extension that $ht P \lt ht P\cap A$

Question

All rings are commutative Noetherian with 1.
Matsumura's Exercise 9.8. Let $A$ be a ring and $A\subset B$ an integral extension. If $P$ is a prime ideal of $B $ with $p = P\cap A$ then $ht P \leq ht p$.
I have examples that equality occurs; i.e. $ht P = ht p$. But I cannot see examples that $ht P \lt ht p$. Can you give such examples, please?
Thank you.