Biomimetics has played a key role in the evolution of artificial neural networks. Thus far, in silico metaphors have been dominated by concepts from neuroscience and cognitive psychology. In this paper we introduce a different type of biomimetic model, one that borrows concepts from the immune system, for designing robust deep neural networks. This immuno-mimetic model leads to a new computational biology framework for robustification of deep neural networks against covariate shifts and adversarial attacks. Within this Immuno-Net framework we define a robust adaptive immune-inspired learning system (Immuno-Net RAILS) that emulates the in silico the adaptive biological mechanisms of B-cells that are used to defend a mammalian host against pathogenic attacks. When applied to image classification tasks on benchmark datasets, we demonstrate that Immuno-net RAILS results in improvement of as much as 12.5% in adversarial accuracy of a baseline method, the DkNN-robustified CNN, without appreciable loss of accuracy on clean data.