  Our work is devoted to the study of Pontryagin's stochastic maximum principle
for a mean-field optimal control problem under Peng's $G$-expectation. The
dynamics of the controlled state process is given by a stochastic differential
equation driven by a $G$-Brownian motion, whose coefficients depend not only on
the control, the controlled state process but also on its law under the
$G$-expectation. Also the associated cost functional is of mean-field type.
Under the assumption of a convex control state space we study the stochastic
maximum principle, which gives a necessary optimality condition for control
processes. Under additional convexity assumptions on the Hamiltonian it is
shown that this necessary condition is also a sufficient one. The main
difficulty which we have to overcome in our work consists in the
differentiation of the $G$-expectation of parameterized random variables. As
particularly delicate it turns out to handle with the $G$-expectation of a
function of the controlled state process inside the running cost of the cost
function. For this we have to study a measurable selection theorem for
set-valued functions whose values are subsets of the representing set of
probability measures for the $G$-expectation.
