Context. The star formation process in large clusters/associations can be strongly influenced by the feedback from high mass stars. Whether the resulting net effect of the feedback is predominantly negative (cloud dispersal) or positive (triggering of star formation due to cloud compression) is still an open question. Aims. The Carina Nebula complex (CNC) represents one of the most massive star-forming regions in our Galaxy. We use our Herschel far-infrared observations to study the properties of the clouds over the entire area of the CNC (with a diameter of ≈ 3.2 • , which corresponds to ≈ 125 pc at the distance of 2.3 kpc). The good angular resolution (10 ′′ − 36 ′′ ) of the Herschel maps corresponds to physical scales of 0.1 -0.4 pc, and allows us to analyze the small-scale (i.e. clump-size) structures of the clouds. Methods. The full extent of the CNC was mapped with PACS and SPIRE in the 70, 160, 250, 350, and 500 µm bands. We determine temperatures and column densities at each point in this maps by modeling the observed far-infrared spectral energy distributions. We also derive a map showing the strength of the UV radiation field. We investigate the relation between the cloud properties and the spatial distribution of the high-mass stars, and compute total cloud masses for different density thresholds. Results. Our Herschel maps resolve, for the first time, the small-scale structure of the dense clouds over the entire spatial extent of the CNC. Several particularly interesting regions, including the prominent pillars south of η Car, are analyzed in detail. We compare the cloud masses derived from the Herschel data to previous mass estimates based on sub-mm and molecular line data. Our maps also reveal a peculiar "wave"-like pattern in the northern part of the Carina Nebula. Finally, we characterize two prominent cloud complexes at the periphery of our Herschel maps, which are probably molecular clouds in the Galactic background. Conclusions. We find that the density and temperature structure of the clouds in most parts of the CNC is dominated by the strong feedback from the numerous massive stars, rather than random turbulence. Comparing the cloud mass and the star formation rate derived for the CNC to other Galactic star forming regions suggests that the CNC is forming stars in an particularly efficient way. We suggest this to be a consequence of triggered star formation by radiative cloud compression.