I know a couple of people that hunt Nutria with PCP's like the Air Force Condor. Nutria are a nuissance in most southern states and I have been reading that they are slowly moving north like in the Carolinas and even Virginia. Most states encourage people to kill them and have no limit or restrictions on Nutria.
You will need to make it a head shot with an airgun and move in close, or use a medium-high powered PCP. Nutria can get large and reach or exceed 20lbs at times. Many people take them with rimfire too.
I 've heard dessert jacks don't taste so great, but I have no personal experience with that.