An important use case of next-generation wireless systems is device-edge co-inference, where a semantic task is partitioned between a device and an edge server. The device carries out data collection and partial processing of the data, while the remote server completes the given task based on information received from the device. It is often required that processing and communication be run as efficiently as possible at the device, while more computing resources are available at the edge. To address such scenarios, we introduce a new system solution, termed neuromorphic wireless device-edge co-inference. According to it, the device runs sensing, processing, and communication units using neuromorphic hardware, while the server employs conventional radio and computing technologies. The proposed system is designed using a transmitter-centric information-theoretic criterion that targets a reduction of the communication overhead, while retaining the most relevant information for the end-to-end semantic task of interest. Numerical results on standard data sets validate the proposed architecture, and a preliminary testbed realization is reported.