Thanks, apparently I can't read!
The orange, nearly-coin-cell-sized device sure looks like a "super-cap" that could backup the rtc (it's also right next to the microcontroller which may have the rtc built-in, but I can't see enough of the board to determine if there's a separate clock chip). It looks like it might be a 5.5 volt, 47 millifarad (0.047F) cap.
Can you get a better shot of the board, particularly showing that orange cap? I couldn't find an image online that was clear enough.
Eaton, Elna, Kemet, Vishay are some manufacturers that make 5.5 volt 0.047F supercaps. They may also be listed as 47mF (milli farads), 47,000uF (micro farads). You can get them at Digikey, Mouser, etc. for a few bucks.