Decentralized learning enables edge users to collaboratively train models by exchanging information via deviceto-device communication, yet prior works have been limited to wireless networks with fixed topologies and reliable workers. In this work, we propose an asynchronous decentralized stochastic gradient descent (DSGD) algorithm, which is robust to the inherent computation and communication failures occurring at the wireless network edge. We theoretically analyze its performance and establish a non-asymptotic convergence guarantee. Experimental results corroborate our analysis, demonstrating the benefits of asynchronicity and outdated gradient information reuse in decentralized learning over unreliable wireless networks.