It used to be more common for a chip company to own their own fab. But over the years that has become less popular.
Today, there is only one company (Intel) which is fully vertically integrated in that they control the architecture, the design, and the fab; and they make some de minimis amount of consumer-level computers too.
Samsung is sort of there, in that they design chips and fab some of them; but not all of them.
AMD used to have a fab, but they spun it out as GlobalFoundries.
As others pointed out, it's more advantageous to have several fabs available as options rather than being stuck with your own in-house fab.
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People in this thread are way way way overblowing the benefits of smaller process nodes. There are certainly benefits, but its not the be-all end-all of processors. It's still entirely possible for a 28nm Intel chip to be far more powerful than a 5nm ARM cortex chip.