Neutrinoless double-β decay (0νββ decay) is a hypothetical process that can occur if the neutrino is its own antiparticle. The COBRA Collaboration operates a demonstrator to search for these decays at the Laboratori Nazionali del Gran Sasso in Italy using CdZnTe semiconductor detectors. The exposure of 234.7 kg d considered in this analysis was collected between September 2011 and February 2015. The analysis focuses on the decay of the nuclides Cd114,Te128,Zn70,Te130, and Cd116. A Bayesian analysis is performed to estimate the signal strength of 0νββ decay. No signal is observed for any of these nuclides. Therefore, the following half-life limits at 90% credibility are set: T1/20ν>1.6×1021yr (Cd114),T1/20ν>1.9×1021yr (Te128),T1/20ν>6.8×1018yr (Zn70),T1/20ν>6.1×1021yr (Te130), and T1/20ν>1.1×1021yr (Cd116).