I'm not advocating using poison. I would try other things first. I had an infestation of rats in one of my coops. It was the first coop built and DH put a ceiling in as well as inside walls around the bottom of the coop. I had been seeing rats from time to time in our barn which is right behind the coop and rat droppings in the coop. I tried many things to no avail so eventually went with poison. When I tore out the ceiling and the walls dozens of rats poured out of all sizes and there were lots of nests that I took out in the ceiling and wall cavities. I bought some Rat bait stations and put them where only the rats could get to the bait in our barn, but as a added precaution to keep other critters away from the bait. I put the bait stations, similar to @aart's suggestion, in something so I used pet carriers so only the rats could get to it. There were tunnels around the coop that I assumed were made by the rats. Surprisingly I didn't find any dead rats laying around. I also assumed they went into their tunnels and died so that is why I didn't find dead ones laying around. The bait stations I bought have a little window above the bait so I can check it daily. Eventually the baits got to where they weren't being touched and I stopped seeing the rats and droppings. I have several game cameras but during that time I didn't see any predators. I do have electric wires around my coops and pens and I believe the predators know it and the adults teach their young that a chicken isn't worth getting zapped for. Good luck...
