We analyze a transboundary pollution differential game where, in addition to the standard temporal dimension, a spatial dimension is introduced to capture the different geographical relationships among regions. Each region behaves strategically and maximizes its welfare net of environmental damage caused by the pollutant stock. The emission-output ratio in each region is reduced by investment in clean technology which is region specific and evolves over time. The spatio-temporal dynamics of the pollutant stock is described by a parabolic partial differential equation. Using aggregate variables for the environmental variables we study the feedback Nash equilibrium of a discrete-space model which could be seen as a space discretization of the continuous-space model. The discrete-space model still presents the three main features of the original formulation: the model is truly dynamic; the decision agents behave strategically; and the model incorporates spatial aspects. For special functional forms previously used in the literature of transboundary pollution dynamic games we analytically characterize the feedback Nash equilibrium and evaluate the impact of the introduction of the spatial dimension in the economic-environmental model. We show that our spatial model is a generalization of the model that disregards the spatial aspects in the sense that the behavior of the environmental variables at the equilibrium in the non-spatial setting can be reproduced as the parameter describing how pollution diffuses among regions tends to infinity and the stocks of pollution in the regions are instantaneously mixed, which is the main hypothesis made in the non-spatial differential game.