The drag-free satellites of LISA will maintain the test masses in geodesic motion over many years with residual accelerations at unprecedented small levels and time delay interferometry (TDI) will keep track of their differential positions at a level of picometers. This may allow investigations of fine details of the gravitational field in the solar system previously inaccessible. In this spirit, we present the concept of a method for measuring directly the gravitational effect of the density of diffuse local dark matter (LDM) with a constellation of a few drag-free satellites, by exploiting how peculiarly it would affect their relative motion. Using as a test-bed an idealized LISA with rigid arms, we find that the separation in time between the test masses is uniquely perturbed by the LDM, so that they acquire a differential breathing mode. Such an LDM signal is related to the LDM density within the orbits and has characteristic spectral components, with amplitudes increasing in time, at various frequencies of the dynamics of the constellation. This is the relevant result in that the LDM signal is brought to non-zero frequencies.