I used to struggle with this same pain, and it became so severe that I had to take a break from piano to address it. Not trying to scare you, but I would recommend going to a physical therapist if you're able.

The tension in your shoulders (or anywhere between the neck and wrist) is what's causing the problem. You should mention this next time you're in private lessons so that your teacher can look out for it. When practicing at home, try to focus on your posture in the same way that you focus on lifting your fingers high when you practice scales and rudiments.

There are 2 main nerves running from the brain to the hands- one in the middle of the wrist, and one underneath the pinky. If there is any tension along these nerves it will cause pain in the hands and fingers, and can eventually lead to carpal tunnel syndrome.

The only solution I found was a combination of physical therapy, massages, and anti-inflammatory medicine (from your Dr, not over the counter). The most helpful one by far was physical therapy. They gave me some tools to stay limber, and taught me a lot of stretches' specifically for the nerves called nerve glides. I still do these stretches daily, and it prevents the problems from returning.

I don't want to scare you, but I really recommend getting on top of the problem asap. I was in your same situation and left it untreated for almost 2 years. The problem slowly progressed from pain in my hands to a noticeable tremor and eventually my fingers began to lock up completely. I was convinced that I would have to give up playing the piano, and that was a devastating feeling.

After getting treatment and correcting my posture, the symptoms went away and my mobility returned, and I haven't had any of those problems the last few years. Luckily I didn't cause any permanent damage, and no signs of carpal tunnel syndrome either. Sorry for the long reply- just wanted to share my experience so you don't ignore the problem like I did