The pressure activated (by your foot) brake lamp switch is notorious for becoming gummed up with dirty brake fluid.
C1AZ-13480-A (replaced 91A-13480) .. Brake Lamp Switch (Motorcraft SW-24) / Available from Ford and a gazillion autoparts stores.
Applications: Myriad 1939/66 FoMoCo vehicles / Same as Studebaker #666858 = 1941/66 Studebaker, 1951/54 Packard, 1955/56 Packard Clipper w/o Torsion Bar suspension, 1957/58 Packabaker.