We consider the Navier-Stokes equations in unbounded domains $\Omega \subseteq \mathbb R ^n$ of uniform $C^{1,1}$-type. We construct mild solutions for initial values in certain extrapolation spaces associated to the Stokes operator on these domains. Here we rely on recent results due to Farwig, Kozono and Sohr, the fact that the Stokes operator has a bounded $H^\infty $-calculus on such domains, and use a general form of Kato's method. We also obtain information on the corresponding pressure term.