Why does the keyless entery system stay on and light blinking?

Hello, first - do you have any non-Lincoln components installed, such as an aftermarket alarm system? Anything at all that has been added? It sounds like a non-factory module is causing some troubles, or wiring that has been modified. As this is a variable, it's the most likely cause of the concern. Beyond that, if everything is perfectly "factory" - then it's something that will have to be checked out by a trained technician equipped with a Ford IDS diagnostic tool (or equivalent) which can monitor component operation. Then circuit testing of the related items that are working unexpectedly. Use this link to find some shops near you that can help with your concern: www.repairpal.com/repair-shops