Obviously it wouldn't be fast / profitable / etc. But you only asked if it
can.
Select CompactFlash cards (that don't use a dedicated controller but a generic uC)
might be able to mine bitcoin, but I'm doubting it. I'm not even sure if you could do it with the more capable ones SD cards. You would certainly have a hard time addressing them, though.
For some points, see:
it turns out that every flash memory disk ships with a reasonably powerful microcontroller [...] The embedded microcontroller is typically a heavily modified 8051 or ARM CPU. In modern implementations, the microcontroller will approach 100 MHz performance levels, and also have several hardware accelerators on-die.