I tried to put a spare battery on with no luck. But after posting this I've seen that not only the configs are not saved but the date/hour are also reset. So... maybe the spare battery is also dead. Ordered a new pack of batteries and it should arrive today. I'll try them and come back here to report.
I agree, this definitely looks like a battery issue. When installing a new battery, check the battery bay. Sometimes those little connectors around it lose their elasticity so you have to bend them a bit to allow for better contact with the battery.