Light transport Captain reported a near miss with another aircraft while setting up to land at a non-towered airport. Reportedly; the other aircraft had done a touch-and-go and was not paying attention to the frequency.

Narrative

In Aircraft X on approach to Runway 23 at AND we had a near midair with Aircraft Y. Situation was we were on left downwind for 23. I was the pilot flying. We heard a jump plane on frequency who said 4 minutes till jumpers away. My second in command talked with him on CTAF and we decided we had time to land before they jumped - although he said they wouldn't interfere with us anyway. She made all appropriate CTAF calls. Fast forward to 2-mile final and we heard jumpers away. We decided to break off the approach and turn left back to the downwind. We wanted to use caution to avoid the jumpers who were expected to land at the far end of the runway off to the right. I expected he would wait until we landed to release his jumpers but; no such luck. We had briefed this possibility and that go-around was no big deal. We didn't want to risk being so close to jumpers under canopy. We ran all checklists and set up to land again making plenty of CTAF calls and coordinating with the jump plane to land before him. About 2-mile final I saw what appeared to be Aircraft Y head on slightly low. It appeared to be climbing as we descended on final. I called 'traffic going around' and started an aggressive climbing left turn. As soon as we got the flaps to 2; the TCAS started screaming descend; descend now; then immediately reversed to climb; climb now. We were already climbing and banking left with traffic in sight so we followed the climb RA and kept the turn going. We passed VERY close to Aircraft Y. I'd estimate 50 - 100 ft. clearance. Aircraft Y then came on frequency say 'sorry to the jet I had my volume down' and he departed to the west. Based on later info from the jump pilot we think Aircraft Y actually did a touch-and-go on Runway 5; right beside the jumpers; and was climbing on upwind when we encountered him. We landed without father incident. Pilot of Aircraft Y was careless and reckless in my opinion.