Kind of a can of worms here. You can get a repeater, and some very high quality ones today, but almost none of them have the power to kill a rat. It's generally considered that about ten foot pounds of muzzle energy is about the minimum for this duty, and the repeaters generally just don't get there.
The exception are some of the pre-charged pneumatics, but they are expensive and require some support gear to make them run. They need to be charged with high pressure air. Most guys use a SCUBA tank and just get it refilled at a dive shop. Kind of a hassle just for some rat control. Hatsan makes a good one, their Velox model.
Single shots are still the way to go for reliable rat killing power. Crossman still makes a dirt cheap, reliable, powerful pump up pistol. It has the power and accuracy you need for rat killing. I'm pretty sure they are the only game in town these days. Pump guns just aren't popular anymore.
Next up would be a spring piston gun. There are a lot of cheap ones out there that just don't have the power either, but there is one really good one. Kind of spendy, but really top drawer quality. The Weihrauch HW 45 is one of my all time favorites. One of the most powerful air pistols (at least self contained ones) you can buy today. Mine is an oddball on .20 (5mm) caliber, but they make .177 and .22 as well. A real rat killer, believe me.
Another option would be an older Sheridan or Benjamin pump pistol. They are essentially the same gun, with the Sheridan in their normal .20 caliber and the Benjamin in either .177 or .22. They can be found on eBay and other such sites fairly cheap these days.
So, while somewhat limited, there are options for a "rat pistol". Not as many as there used to be, some fairly expensive examples, but at least one cheap and proven option.