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Automatic pathological speech detection relies on deep learning (DL), showing promising performance for various pathologies. Despite the critical importance of robustness in healthcare applications like pathological speech detection, the sensitivity of DL-based pathological speech detection approaches to adversarial attacks remains unexplored. This paper explores the impact of acoustically imperceptible adversarial perturbations on DL-based pathological speech detection. Imperceptibility of perturbations, generated using the projected gradient descent algorithm, is evaluated using speech enhancement metrics. Results reveal a high vulnerability of DL-based pathological speech detection to adversarial perturbations, with adversarial training ineffective in enhancing robustness. Analysis of the perturbations provide insights into the speech components that the approaches attend to. These findings highlight the need for research in robust pathological speech detection.