I think the faceplate is not the plastic part with the lines running horizontally on it, but the chrome bezel that surrounds the faceplate.
Actually on the DIN chassis of the radio, there are 4 small springs that are fastened to the chassis and slip into small slots on the bezel. Possibly they are weak from age. Try to pull the chassis out from the dash and bend the small spring clip tabs back a bit.
Hope this helps.
Walter Klatt
1963, 1965, 1967 230SL's