  We study the well-posedness of a system of forward-backward stochastic
differential equations (FBSDEs) corresponding to a degenerate mean field type
control problem, when the diffusion coefficient depends on the state together
with its measure and also the control. Degenerate mean field type control
problems are rarely studied in the literature. Our method is based on a lifting
approach which embeds the control problem and the associated FBSDEs in
Wasserstein spaces into certain Hilbert spaces. We use a continuation method to
establish the solvability of the FBSDEs and that of the G\^ateaux derivatives
of this FBSDEs. We then explore the regularity of the value function in time
and in measure argument, and we also show that it is the unique classical
solution of the associated Bellman equation. We also study the higher
regularity of the linear functional derivative of the value function, by then,
we obtain the classical solution of the mean field type master equation.
