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Actually, you can. It’s just a more involved and longer process than center fire ammo. If you do some searching online, you can find the company that sells the dies. There’s a few videos and articles on how to go about it.
I've studied this topic a bit. From what I can tell, a group of reloaders really threw a lot of money (and even more time) at this concept some years ago, but achieved very poor accuracy from their reloaded ammo. Applying the new priming mixture consistently seems to be the stumbling block.
If you really need something in the .22LR power range, it'd be far easier to use a small centerfire like a .22 Hornet and tinker with light loads.
.38 wadcutter rounds (reloaded, cast your own bullets) are only a bit more than .22 power, very accurate, lots of great loads developed for superb accuracy. Cost is a bit lower than .22 now (depends on the price of primers), but mild, accurate, great shooting stuff. Out of a rifle, quieter than .22 hi vel.