The Atmega2560V can run at that low voltage
ATmega640V/ATmega1280V/ATmega1281V: • 0 - 4MHz @ 1.8V - 5.5V, 0 - 8MHz @ 2.7V - 5.5V
The Arduino Mega uses the nonV chip and needs 4.5V no matter the speed, see the lower right graph.
How small of a board are you after? This board I offer could be assembled with a 2560V and 4 MHz crystal. Do you have a 1.8V power source?
http://www.crossroadsfencing.com/BobuinoRev17/