Along with the successful deployment of deep neural networks in several application domains, the need to unravel the black-box nature of these networks has seen a significant increase recently. Several methods have been introduced to provide insight into the inference process of deep neural networks. However, most of these explainability methods have been shown to be brittle in the face of adversarial perturbations of their inputs in the image and generic textual domain. In this work we show that this phenomenon extends to specific and important high stakes domains like biomedical datasets. In particular, we observe that the robustness of explanations should be characterized in terms of the accuracy of the explanation in linking a model's inputs and its decisions - faithfulness - and its relevance from the perspective of domain experts - plausibility. This is crucial to prevent explanations that are inaccurate but still look convincing in the context of the domain at hand. To this end, we show how to adapt current attribution robustness estimation methods to a given domain, so as to take into account domain-specific plausibility. This results in our DomainAdaptiveAREstimator (DARE) attribution robustness estimator allowing us to properly characterize the domain-specific robustness of faithful explanations. Next, we provide two methods, adversarial training and FAR training, to mitigate the brittleness characterized by DARE, allowing us to train networks that display robust attributions. Finally, we empirically validate our methods with extensive experiments on three established biomedical benchmarks.