We consider a pair of domains Ω b and Ω s in ℝ n and we assume that the closure of Ω b does not intersect the closure of εΩ s for ε ∈] 0, ε0[. Then for a fixed ε ∈] 0, ε0 [we consider a boundary value problem in ℝ n ∖(Ω b ∪ εΩ s ) which describes the steady state Stokes flow of an incompressible viscous fluid past a body occupying the domain Ω b and past a small impurity occupying the domain εΩ s . The unknowns of the problem are the velocity field u and the pressure field p, and we impose the value of the velocity field u on the boundary both of the body and of the impurity. We assume that the boundary velocity on the impurity displays an arbitrarily strong singularity when ε tends to 0. The goal is to understand the behaviour of (u, p) for ε small and positive. The methods developed aim at representing the limiting behaviour in terms of analytic maps and possibly singular but completely known functions of ε, such as ε−1, log ε.