Neuromorphic computing has become popular in robotics, edge devices and IoT because of its energy efficiency and biological inspiration. These systems are based on spiking neural networks (SNNs), which process information in discrete spike events, providing real-time and low-power operation. However, in spite of these advantages, the safety of spiking neuromorphic systems has not been studied extensively as compared to traditional deep learning systems. In this paper, we present NeuroMimicry Attacks, a type of adversarial evasion attack in which adversarial examples are patterns of the spike-train that are highly similar to a legitimate activity but reach malicious goals. These attacks take advantage of the temporal and spatiotemporal properties of SNNs and are challenging to identify using the current anomaly detection systems. This work has four contributions: first, a taxonomy of mimicry-based adversarial attacks is created; second, algorithms to generate realistic spike-train perturbations and synthetic mimicry patterns are proposed; third, defense strategies are proposed, including spatiotemporal anomaly detection and adversarial training; fourth, the work has been experimentally validated using benchmark neuromorphic datasets and platforms. Findings indicate that the NeuroMimicry Attack is a major threat and requires strong defensive systems specific to neuromorphic systems.