Battery light stays on
The battery light on the instrument panel stays on after the engine has started
The alternator may be bad or it could have a wiring issue. Have the alternator tested. If you are handy and have a volt meter, check the voltage at the battery with the engine running. It should be 13.5 to 14.7 volts. If it is lower than 13 the charging system is not working properly and the system needs to be evaluated.
On this model F150. 13.2 or low is a bad alternator. You need to load test both the battery and the alternator. Touch the battery terminals and watch closely as someone else starts the vehicle. If the voltage goes below 10V, you need a new battery. Then while the vehicle is running, touch the terminals again and see what voltage is. 13.3 - 14.8 is specified as a good motorcraft standard alternator. Next turn on some high drain accessories like bright lights, blower and radio and see what the low is on the volt meter. 13.2 or below, replace the alternator. Just had 2 different males in my shop that had bad alternators at 13.2 volts. Hope this helps.