Return it Sony should give you a free replacement.

If it doesn't work while it's plugged in either then it is not the battery.

Sounds like it's broken.

Unusual that cases like this are so rarely reported, even with 3 year old controllers in circulation.

If it were me, I'd disassemble the entire controller, clean out all components and then check the main circuit board by plugging it into a powered USB port to see if the LED power lights turn on.

It's possible something simply came disconnected (controller has several plug-in connectors), or it could be a catastrophic failure (broken circuit board or damaged chip component). Everything has a finite life cycle.

I've seen one controller get stepped on (cracked case) that still works fine, and another that took an impact hard enough to crack the casing that still works fine, so they're pretty durable.