A quantitative analysis is presented of the increasing degree of dynamical correlations with decreasing temperature in a fragile glass-former consisting of a two-dimensional binary mixture of soft discs with a diameter ratio of . The analysis involves a study of the spatial distribution of local relaxation times, defined as the time taken for each particle to first travel a distance r from its initial position. For the binary mixture, a temperature-independent optimum value of is found to maximize the spatial segregation of particles into different kinetic domains. The regions of `fast' and `slow' particles grow in size as the system is cooled, indicating an increasing degree of cooperativity in the particle dynamics. A measure of the linear dimensions of these clusters is provided. It is shown that only `slow' subsets of particles are caged on intermediate timescales and that the lifetime of these slow domains increases dramatically with decreasing temperature in the supercooled mixture. A substantial decay in the incoherent scattering functions can still be accomplished, however, on these timescales, despite the relative immobility of a significant fraction of the system. A further observation is a change in the manner in which relaxation progresses throughout the system with cooling. At high temperatures the initially fast relaxing sites are randomly distributed throughout the system, whereas at low temperatures they tend to be clumped together. This subsequently results in a less homogeneous progression of relaxation at the lower temperatures, since relaxation proceeds primarily by radiating outwards from existing fast centres.