For info, I have the same issue, and was just chatting to an online agent.
They said to me that the Surface Pro 4 was discontinued for support on the 3rd of this month (!), so no spare parts / replacement devices anymore, so there was nothing they could do.
But they also said that I probably shouldn't use it unless I really have to, as the battery could leak or explode, and to try and keep it in a cool place, and to avoid using it with the power supply attached (which is problematic, as the device also suffers from the other SP4 problem whereby it completely discharges itself in a few days, even when it's completely turned off!).
Sounds like the device is dead... Shame, as it was an expensive purchase. Not sure I'll be buying another one either.