Any other issues with the touchpad on the XPS 9560? Mine has started to fail and I lifting above the case a bit. I read that some 9550s has battery expansion issues that created similar problems.
If you see the touchpad lifting then it is indeed a battery issue. First thing is to remove the battery as soon as you possibly can. It is a safety issue and at the very least could damage the laptop to the point where you would be looking at replacing the entire system. Dispose of the battery in a safe environmentally safe way. Then contact Dell parts in your region to purchase a new battery.