CSMA/CA is a popular random-access algorithm for wireless networks, but its stability properties are poorly understood. We consider a linear multi-hop network of three nodes where the neighbouring nodes interfere with each other and medium access is governed by the CSMA/CA algorithm. We assume that the source node is saturated and packets are forwarded through the network, each node transmitting towards its neighbour on the right. We demonstrate that the queue of the second node is saturated (unstable) and the queue of the third node is stable; this confirms heuristic arguments and simulation results found in the research literature. Providing a rigorous proof for the (in)stability of these nodes is complicated by the fact that neither queue is Markovian when considered in isolation, and the two queues are dependent. We then compute the limiting behavior of node 3, and use this to determine the end-to-end throughput of the network. Finally, we vary the access probabilities of the nodes, and evaluate how this affects the stability and throughput of the system.