Neuromorphic computing mimics computational principles of the brain in $\textit{silico}$ and motivates research into event-based vision and spiking neural networks (SNNs). Event cameras (ECs) exclusively capture local intensity changes and offer superior power consumption, response latencies, and dynamic ranges. SNNs replicate biological neuronal dynamics and have demonstrated potential as alternatives to conventional artificial neural networks (ANNs), such as in reducing energy expenditure and inference time in visual classification. Nevertheless, these novel paradigms remain scarcely explored outside the domain of aerial robots. To investigate the utility of brain-inspired sensing and data processing, we developed a neuromorphic approach to obstacle avoidance on a camera-equipped manipulator. Our approach adapts high-level trajectory plans with reactive maneuvers by processing emulated event data in a convolutional SNN, decoding neural activations into avoidance motions, and adjusting plans using a dynamic motion primitive. We conducted experiments with a Kinova Gen3 arm performing simple reaching tasks that involve obstacles in sets of distinct task scenarios and in comparison to a non-adaptive baseline. Our neuromorphic approach facilitated reliable avoidance of imminent collisions in simulated and real-world experiments, where the baseline consistently failed. Trajectory adaptations had low impacts on safety and predictability criteria. Among the notable SNN properties were the correlation of computations with the magnitude of perceived motions and a robustness to different event emulation methods. Tests with a DAVIS346 EC showed similar performance, validating our experimental event emulation. Our results motivate incorporating SNN learning, utilizing neuromorphic processors, and further exploring the potential of neuromorphic methods.