I've got a good air rifle too (Beeman R77 if it matters), but out of all the weapons at my disposal, that would be the last one I'd pick up to use on a mink, weasel, skunk, possum or other. At least one that was not in a trap and running around on the loose.
Even a dinky little Mossberg .410 pump shotgun you can buy at Walmart.....which holds 4 or 5 shells and burps out a lethal dose of lead at close quarters. About 5 to 20X the single pellet an air rifle would dispense and at a much higher velocity.
If you can get close enough to that mink to take those kind of photos he is close enough. A dead center body shot with even a .410 will leave him graveyard dead.