Have yet to encounter one of the "super rats" described here.
I find most claims of animal toughness to really boil down to nothing more than shots that fail to do significant damage to the CNS system. Like the above described tough gut-shot rat, well you can shoot a lot of animals and people in the guts and they will run. Why? Nothing immediately vital there.
I have read descriptions of some deer being said to be tough, plus countless descriptions of hogs, coyotes, opossums, raccoons, and mountain lions said to be tough. Again, it almost always comes down to shots not doing the damage that they need to do and that usually comes down to the animals not being shot in the right places. The animals may die, just not on the spot where they were shot. A hog shot through the heart may run 75-100 yards before the brain runs out of oxygen. This isn't an issue of being tough. The shot to the heart won't necessarily result in the hog dropping dead at the point of impact and failure to take out its means of locomotion (legs and girdles) means the hog can still run, for 10-20 seconds, and cover a lot of distance.
As for needing hollowpoint ammo because these rounds are like FMJ and make "tiny holes," you have to realize the tiny hole is HUGE relative to the size of a rat. Think about it this way. A .22 lr bullet is about the same diameter as a rat's wrist. Imagine a hole about the size of your wrist going through your body. Is that a big hole? Sure, and it is many many times the size of what would be produced by a 9mm bullet, right?
Or, on average rat weighs about a half pound, or 7000 grains. It is being shot with a 40 grain bullet that is 1/175th of the rat's weight. Compare that with a 9mm 115 gr. bullet being shot at about the same velocity at an adult human of 180 lbs or 2,520,000 grains, the bullet being 1/21913th of the human's weight.
A 40 gr. round nose bullet should be more than adequate for killing a rat, without expanding or fragmenting, if the shooting does his/her part.
Airgun folks are regularly killing rats with the same or smaller caliber pellets that are lighter in weight and likely often lower in velocity than what the OP is shooting. They make good shots and put rats down.