The airbags in your 2006 Ford F-150 are part of the supplemental restraint system. The airbag warning light may flash or stay lit continuously when an SRS error is detected. Diagnostic trouble codes associated with the SRS stay in memory for 128 key cycles after the last error detection. Once the malfunction is repaired, depower and repower the system to turn off the light.
Depower the SRS
Turn off all the accessories in your F-150 and turn off the ignition. Pull the red 10A mini-fuse from slot No. 19 in the fuse panel below the passenger side of the dashboard. Turn the ignition key on and watch the airbag warning light. It should come on and stay on continuously without flashing. This verifies that you pulled the correct fuse. Turn the ignition switch off and then disconnect the negative battery cable. Leave the battery cable disconnected for at least a minute to deplete the system's backup power supply.
Repower the SRS
Turn the ignition switch on and then reinstall the fuse into the fuse panel. Connect the negative battery cable. As a precaution, don't allow anyone to sit in the driver's seat while you connect the battery cable. Turn the ignition switch off, wait 10 seconds and then turn it back on. The airbag light should illuminate for six seconds and then turn off. If the light doesn't come on, if it lights continuously, or if it flashes, there is still a malfunction present in the SRS. It takes about 30 seconds for the self-test to complete, so watch the light for at least that long before you claim victory. Use a scan tool to clear DTCs from memory.