Neuromorphic computing is an emerging technology that support event-driven data processing for applications requiring efficient online inference and/or control. Recent work has introduced the concept of neuromorphic communications, whereby neuromorphic computing is integrated with impulse radio (IR) transmission to implement low-energy and low-latency remote inference in wireless Internet-of-Things (IoT) networks. In this paper, we introduce neuromorphic integrated sensing and communications (N-ISAC), a novel solution that enables efficient online data decoding and radar sensing. N-ISAC leverages a common IR waveform for the dual purpose of conveying digital information and of detecting the presence or absence of a radar target. A spiking neural network (SNN) is deployed at the receiver to decode digital data and to detect the radar target using directly the received signal. The SNN operation is optimized by balancing performance metrics for data communications and radar sensing, highlighting synergies and trade-offs between the two applications.