Jeff Liebermann says...

Yep. It's probably the battery. However, if both BP40
battery packs are the same age, it's likely the working
battery pack will soon fail. Buy a replacement BP40 for
both radios.

I did that. One of the big suppliers of generic BP40s also
runs auctions from time to time, with a starting bid of $.01
plus $5.39 shipping for a pair. One of those was ending as
I was about to buy, so I bid a penny and won the auction.
So $5.40 for two BP40s. They claim to be 700 mAh, but of
course there's no way to know about that for sure. But I
think not a bad gamble overall.

If this were my radio, I would buy 4ea AAA NiMH LSD (low
self discharge) cells, such as Eneloop cells, and weld
them into a replacement battery pack. The ability to
charge the battery and know that it will be mostly at
full charge months later, is worthwhile.
https://en.wikipedia.org/wiki/Eneloop
Be sure to check the charge current to make sure that
you're not overcharging or quick charging the Eneloop
batteries.

I've always used Eneloops in my speedlights, and am a big
fan. But the charging setup here is very strange. The
charging cradle appears to put out a minimum of 7V to each
radio, sometimes 12.5V, and it loods like the BP40 has some
smarts built into it that actually controls the charging.
So if I used Eneloops, I would have to take them out of the
radios to charge them. So for the moment I'm going to stick
with the BP40s and use the charging cradle.

The charging contacts on the BP40 are not directly connected
to the outside battery terminals of the 4-pack. So there's
some circuitry in there. When the new ones arrive, I may
take the old bad one apart and see what's in there besides
the cells.