Did you charge the battery overnight when you got it (or at least leave it in the charger for around 8 hours).
I'd suggest doing that with a new battery. Then, remove it and plug it back in again, and let it charge until the light changes again. If you remove a brand new battery before it's fully charged, sometimes it can fool the microprocessor that keeps track of battery charge remaining so that it's giving you a false reading.
Of course, it could also be a defective battery causing the issues.
If you have a store nearby with demo models using the same battery, you may want to ask them if you could try one of their batteries in it. That way, you'd narrow the problem down to the camera or the battery.