RAM memory looses its contents when it's powered off. However, this
natural erasure process is not instantaneous: it can take several
seconds, or even minutes. It has been demonstrated that cooling down a
memory chip can extend its data retention time to hours, which enables
a type of attack known as cold boot
attack.
This type of attack has been studied mostly on DRAM computer modules.
The Arduino memory, however, is based on SRAM, which is physically very
different from DRAM (a transistor feedback loop vs. a capacitor). There
seems to be very little research on cold boot attacks against SRAM. I
found nonetheless a technical report from the computer laboratory of the
University of Cambridge: Low temperature data remanence in static
RAM, by Sergei
Skorobogatov. Quoting from the conclusion:
Contrary to the established wisdom, there are several chips that
retain data for dangerous periods of time at temperatures above −20°C.
The temperature at which 80% of the data are retained for one minute
varies widely between devices. Some require cooling to at least −50°C,
while others retain data for this period at room temperature.
Retention times can be significantly reduced by shorting VCC to ground
rather than by leaving it floating.
From this perspective I would say that, although the attack seems non
obvious, your secret key is not completely safe even in SRAM. Since the
data retention time varies widely between devices, it could be worth
testing on your Arduino. If possible, I would suggest you overwrite the
key after the signing operation.