The article deals with the homogeneous Stokes resolvent system in a 3D exterior domain, under inhomogeneous Dirichet boundary conditions. Solutions to this boundary value problem are estimated in L^p-norms, with the bounds in these estimates depending on the absolute value of the resolvent parameter λ in an explicit way. Two types of boundary data are considered, that is, L^p-and W^{2−1/p, p}-data. It is shown in particular that the L^p-norm of the velocity outside a vicinity of the boundary, after subtraction of the gradient of a certain harmonic function, is bounded by a constant times |λ| \| b\|p. This estimate carries over to the Oseen resolvent system , leading to a result which has applications in the theory of spatial asymptotics of solutions to the 3D time-dependent Navier-Stokes system with Oseen term.