ARM CPU burning hot within 30s after voltage regulator failure
Check the data sheet section 6.2 Absolute Maximum Ratings. Either the Vdd-Vss rating of 4.0V or the "Input voltage on any other pin" rating has been violated, causing permanent damage to the device.

If you sent this board to a Failure Analysis lab, they could etch a hole into the epoxy that encapsulates the IC, and probably show you a photograph of where the overvoltage condition caused metal migration or other permanent damage to the IC. This isn't fixable. What the FA lab would be able to tell you basically is, it looks like someone applied too much voltage -- and you already know that.

As Leon Heller commented, the board should be scrapped and replaced. Not only the ARM itself, but very likely most of the other ICs will have been damaged. Even if you tried to work around the overheating problem, there's no longer any guarantee that IC still behaves like an ARM chip. What if its ALU sometimes makes incorrect results or some of its registers sometimes fail to store data. This board is no longer reliable and should be scrapped.

For what it's worth, mistakes like this do happen -- one of the hazards of working on bare prototype PCB instead of nicely packaged finished consumer goods. In this case it should be easy to order a replacement.