You'll probably get a more informative answer if you ask on a Ford truck forum. I, personally, don't have experience on Ford trucks as new as your rig. My truck is a '91 F250, and its radio has a tape player in it and turns right off with the key.
Knowing nothing about your truck, but with my experience as a test engineer, you probably have a faulty electronics... perhaps a micro-relay that switches to ON as its control voltage floats after shutdown, perhaps because of a bad capacitor or some other component failure. If so, the fault is inside the AV/Nav unit, not out where you can change a component and fix it. There's not much you can do to diagnose it beyond taking it to someone who can open it up, test its test points, compare those values to the values in the diagnostic schematics from the factory.
On the other hand, if you're an electronic technician or electrical engineer, and you have the schematics, then you already know how to go about fixing this. Since your question kinda indicates you're not quite in that select skillset, take the car to the dealer and be prepared to be told you'll have to replace the unit. I bet they don't have any smaller component to replace. Warranties are good on modern cars and trucks, and I hope this is covered, because it may cost several hundred dollars.
Good luck,
Scott
