Diffusion adaptive networks tasked with solving estimation problems have attracted attention in recent years due to their reliability, scalability, resource efficiency, and resilience to node and link failure. Diffusion adaptation strategies that are based on the least-mean-squares algorithm can be nonrobust against impulsive noise corrupting the measurements. Impulsive noise can degrade stability and steady-state performance, leading to unreliable estimates. In previous work [“Robust adaptation in impulsive noise,” IEEE Trans. Signal Process., vol. 64, no. 11, pp. 2851-2865, Jun. 2016], a robust adaptive algorithm for stand-alone agents was developed, one that semi-parametrically estimates the optimal error nonlinearity jointly with the parameter of interest. Prior knowledge of the impulsive noise distribution was not assumed. In this paper, we extend the framework to solve the problem of robust distributed estimation by a network of agents. Challenges arise due to the coupling among the agents and the distributed nature of the problem. The resulting diffusion strategy is analyzed and its performance illustrated by numerical simulations.