A \(C^1\) submanifold \(M\) of \(\mathbb C ^n\) is called generic if at each point \(z\in M\) its real tangent space \(T_z M\) satisfies \(T_z M + J T_z M=\mathbb C ^n\), where \(J\) denotes the standard complex structure of \(\mathbb C ^n\). The authors prove that the pluricomplex Green function (or Siciak’s extremal function) of a \(C^2\) smooth generic compact submanifold (without boundary) of \(\mathbb C ^n\) is Lipschitz continuous. Such a result is relevent in approximation theory and complex dynamics. The main technical tool in the proof is a construction, which goes back to Bishop, of a family of analytic discs attached to the manifold. This method was introduced by the authors in the context of the study of the pluricomplex Green function.