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Top-down feedback in cortex is critical for guiding sensory processing, which has prominently been formalized in the theory of hierarchical predictive coding (hPC). However, experimental evidence for error units, which are central to the theory, is inconclusive, and it remains unclear how hPC can be implemented with spiking neurons. To address this, we connect hPC to existing work on efficient coding in balanced networks with lateral inhibition, and predictive computation at apical dendrites. Together, this work points to an efficient implementation of hPC with spiking neurons, where prediction errors are computed not in separate units, but locally in dendritic compartments. The implied model shows a remarkable correspondence to experimentally observed cortical connectivity patterns, plasticity and dynamics, and at the same time can explain hallmarks of predictive processing, such as mismatch responses, in cortex. We thus propose dendritic predictive coding as one of the main organizational principles of cortex.