Any way to permanently seal natural stone slate tile floor?
There are long-term penetrating sealers that tend to last longer than the usual water-based sealants, but they generally do not create a top-coat.  Look for "penetrating" or "impregnating" sealers.

If you're trying to add a topical sealant (like urethane on wood), those also exist.  I believe they're usually wax-based, though, and I'm not sure how long they would last before you need to recoat.  Probably every couple of years, like fixer1234 said.  For those products, you would look for a "sealer and finish".  You may also be able to use something like a masonry sealer for concrete, but I would check product packaging beforehand.