Yes, there is an internal battery inside that voicemail system.
Exsmogger also mentioned that the R1 is not Y2K compliant.
If the battery needs to be changed out, the system's date will default to the 1980s.
As exsmogger mentioned, it might be a good idea to power the phone system off of a UPS to minimize the effects of commercial-power outages.
My thought would be to see if any other type of Avaya voicemail system [Partner Mail VS or Avaya partner Messaging] will work with the existing system and upgrade.