STM32F411 Nucleo board – what the ferrite bead is used for

How are you transferring the data into the FSMC address space? Are you manually copying it from RAM? Constructing it on the fly and writing it to the address space?

To transfer it in the minimum possible time and without CPU intervention, I would:

1. Reserve the 48kB of RAM necessary to store a complete LCD frame (a "frame buffer").
2. Use DMA in a continuous loop to transfer the frame buffer into the FSMC address space.
3. Manually update the contents of the frame buffer as needed; DMA/CPU contention will sort itself out.

The biggest problem with this approach is that you may not have 48kB of RAM spare, given that the STM32F1 has at most 64kB of RAM.

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.