Battery light on dashboard will not go off.

OH LOOK CLAUDE A FIRE UNDER THE HOOD!

YES MAUDE BUT NO RED LIGHT ON DASH. NOT TO WORRY!

Light on key off is a parasitic drain. Many regs will switch ON and drain batter. The moniker IDIOT LIGHT really fits.

I alway carry some electrical tape, small piece fixes those lights every time.

Is it dim or bright when still on? The battery dash light in our 14 yr old truck is always on dimly since the truck was new. It goes bright at start then drops to dim when truck starts. It went bright when we did have a real battery problem one time. On second alternator and still like that. It must be some "leakage" thing as mentioned above.
Mex went through the four wire plug in the alternator a couple years ago with me on this too.

So IMO the OP needs to worry if the light stays bright but not if it acts properly but just stays dim instead of going right out.

I'll bet it doesn't matter...

I'll bet it's a GM chassis, notorious for bad ignition switches, key is off but some of the circuits are still live.

Tends to happen when someone asks a question and doesn't get the magical answer they were hoping for.

One OP post 06/06 then POOF into thin air...

Unplug reg at alternator. What happens?

Different years different makes demand different answers...