Do your research before you act. If you have a local Animal Control, see what they suggest or how hard they laugh. Ours charges $20 for every live trapped animal you bring in. Too expen$ive for me.
Contact your local wildlife rescue center, if you have one. They may have a better place to put/keep the mink. Don't know if they're too scarce in your area, this one may have a purpose in a wilder place away from chickens.
Check the local laws on shooting a gun in your yard. I live in an acreage community but my pal got in trouble for shooting a raccoon that ate a nest of chicks in my yard. It was illegal to discharge her gun in a residential neighborhood, which is what we technically are. My neighbors, who work for the sheriff's ofc, heard the shots & called their boss.
I don't think drowning is a bad idea for animals like raccoons & rats whose numbers are excessive. But I first *carefully* wire the live trap securely shut, otherwise the doors can bend or loosen enough for the animal to escape. Then I fill the barrel first so the animal doesn't have to panic waiting while the water rises above its head. Within a minute it's over.
They make good fertilizer buried deep on the drip line of a tree, but yours may make a better muff.