Can secp256k1 be implemented efficiently on a 8-bit processor and very limited RAM? We are trying to build a much simpler version of Trezor. We do have an finished PCB design from a previous project and would like to know whether this design can be modified, without changing the processor, to sign Bitcoin transactions. The current processor is a TI CC2540. Is it possible to get a minimalistic secp256k1 implementation to run on it, or are we just out of luck?
I don't see why not.  Most of what you need to do is just modular arithmetic with big integers (around 2^256).  There's no obstruction to doing this on an 8-bit microprocessor, it just takes a few more instructions.  You might even find some existing code for arbitrary-precision arithmetic.  256 bits is 32 bytes, so 8K of memory seems like plenty.