It’s one of those things that’s effectively impossible to determine. A lot of modern breeds are descended from ancient landraces, but it’s hard to pinpoint the moment people took up selective breeding and began developing them into the breed.
Some of the breeds known to be old include the Egyptian Mau, Persian (at one point, Persians were simply *any* long haired cat in the fancy), Japanese Bobtail, Chartreux, Korat, and the original Palace Cats, the ancestors of Thai cats and modern type Siamese/Orientals.
I would say probably Egyptian cats most likely, the Egyptian Mau and perhaps earlier versions were known to be mummified starting I think from perhaps as early as 10,000 BCE. Cats domesticated themselves over a period of several thousand years by living near human agricultural centers where they helped keep the rodent population under control. Siamese and forest cats are also very old breeds started from domesticated cats naturally in the area. The first wiki link has a paragraph which describes a history of early cat domestication in various areas.