We investigate reliability and component importance in spatially distributed infrastructure networks subject to hazards characterized by large-scale spatial dependencies. In particular, we consider a selected IEEE benchmark power transmission system. A generic hazard model is formulated through a random field with continuously scalable spatial autocorrelation to study extrinsic common-cause-failure events such as storms or earthquakes. Network performance is described by a topological model, which accounts for cascading failures due to load redistribution after initial triggering events. Network reliability is then quantified in terms of the decrease in network efficiency and number of lost lines. Selected importance measures are calculated to rank single components according to their influence on the overall system reliability. This enables the identification of network components that have the strongest effect on system reliability. We thereby propose to distinguish component importance related to initial (triggering) failures and component importance related to cascading failures. Numerical investigations are performed for varying correlation lengths of the random field to represent different hazard characteristics. Results indicate that the spatial correlation has a discernible influence on the system reliability and component importance measures, while the component rankings are only mildly affected by the spatial correlation. We also find that the proposed component importance measures provide an efficient basis for planning network improvements.