Mr. Shepherd uses a hyperbole in Chapter 3 when he tries to persuade Sir Elliot to let Kellynch Hall to the Crofts, a married Navy couple without children:
A house was never taken good care of, Mr Shepherd observed, without a lady: he did not know, whether furniture might not be in danger of suffering as much where there was no lady, as where there were many children. A lady, without a family, was the very best preserver of furniture in the world.
In trying to make a convincing argument, Mr. Shepherd makes the definitive statement that a childless woman is best-suited for (what he sees as) important tasks like preserving furniture and keeping a home. Although Mr. Shepherd exaggerates by making the statement that childless women are the best "preserver[s] of furniture in the world," his sentiment reflects the implied gender roles and societal expectations of the time. Women were barred from owning property, and marriage was not only an expected social norm, but also a way to elevate one’s security, status, or social rank. Moreover, women were often relegated to the realm of the domestic, regardless of whether or not they were interested in such a life. Mr. Shepherd's hyperbolic statement that a woman without a family is the best keeper of furniture in the entire world is therefore in keeping with the time period's societal expectations surrounding women and housework.
In Chapter 15, Sir Walter Elliot uses a hyperbole as he complains to Anne about the unattractiveness of Bath's women:
The worst of Bath was the number of its plain women. He did not mean to say that there were no pretty women, but the number of the plain was out of all proportion. He had frequently observed, as he walked, that one handsome face would be followed by thirty, or five-and-thirty frights; and once, as he had stood in a shop on Bond Street, he had counted eighty-seven women go by, one after another, without there being a tolerable face among them [...] and as for the men! They were infinitely worse. Such scarecrows as the streets were full of.
Sir Walter’s statements about the occupants of Bath are of course not literal. He exaggerates the number of “unattractive” women he sees and even compares the men of Bath to scarecrows. The ridiculousness of Sir Walter’s commentary emphasizes just how extremely vain he is, a character trait that Austen satirizes and critiques throughout the novel. His point in this moment also underscores a certain pretentious elitism, as he goes out of his way to frame Bath as the kind of place only suitable for unattractive people—an obvious exaggeration that highlights his superficiality and his (perhaps undeserved) sense of superiority over the people around him.
Austen occasionally uses hyperbole in Persuasion to poke fun of a character’s weakness or moral failings. In Chapter 18, for instance, Mary Elliot Musgrove uses a hyperbole in a letter she writes to Anne:
I am sorry to say that I am far from well; and Jemima has told me the butcher says there is a bad sore-throat very much about. I dare say I shall catch it; And my sore-throats, you know, are always worse than anybody’s.
Mary, of course, exaggerates in her letter to Anne, dramatizing her illness. She makes an extreme (and unfounded) claim that her sore throats are worse than anybody else’s. This exaggeration for the sake of emphasis is an instance of hyperbole. This exaggeration is intentional—in doing so, Austen highlights Mary Elliot’s tendency to complain, a recurring negative trait of Mary's.