You're gonna most likely need a new alternator and maybe battery as well. It sounds like the alternator is still working to some extent, but not charging enough to keep the battery up to charge. If the light is on, it means the alternator is bad.
Good way to check is to run a voltmeter across the battery while the car is running. Running and charging should be between ~13.5-14v, fully charged battery with the car off should read about 12v. If it's making <13v with the car on, alternator is bad.
2002 Carbon Black ///M3 - current
2002 Stahlgrau ///M3 - RIP
2001 325i - sold