Context. Using the Hamilton Échelle Spectrograph at Lick Observatory, we have obtained precise radial velocities (RVs) of a sample of 373 G- and K-giant stars over more than 12 yr, leading to the discovery of several single and multiple planetary systems. The RVs of the long-period (~53 yr) spectroscopic binary ε Cyg (HIP 102488) are found to exhibit additional regular variations with a much shorter period (~291 days). Aims. We intend to improve the orbital solution of the ε Cyg system and attempt to identify the cause of the nearly periodic shorter period variations, which might be due to an additional substellar companion. Methods. We used precise RV measurements of the K-giant star ε Cyg from Lick Observatory, in combination with a large set of RVs collected more recently with the SONG telescope, as well as archival data sets. We fit Keplerian and fully dynamical N-body models to the RVs in order to explore the properties of a previously known spectroscopic stellar companion and to investigate whether there is an additional planetary companion in the system. To search for long-term stable regions in the parameter space around the orbit of this putative planet, we ran a stability analysis using an N-body code. Furthermore, we explored the possibility of co-orbital bodies to the planet with a demodulation technique. We tested the hypothesis of ε Cyg being a hierarchical stellar triple by using a modified version of the N-body code. Alternative causes for the observed RV variations, such as stellar spots and oscillations, were examined by analyzing photometric data of the system and by comparing its properties to known variable stars with long secondary periods and heartbeat stars from the literature. Results. Our Keplerian model characterizes the orbit of the spectroscopic binary to higher precision than achieved previously, resulting in a semi-major axis of a = 15.8 AU, an eccentricity of e = 0.93, and a minimum mass of the secondary of msini = 0.265 M_⊙. Additional short-period RV variations closely resemble the signal of a Jupiter-mass planet orbiting the evolved primary component with a period of 291 d, but the period and amplitude of the putative orbit change strongly over time. Furthermore, in our stability analysis of the system, no stable orbits could be found in a large region around the best fit. Both of these findings deem a planetary cause of the RV variations unlikely. Most of the investigated alternative scenarios also fail to explain the observed variability convincingly. Due to its very eccentric binary orbit, it seems possible, however, that ε Cyg could be an extreme example of a heartbeat system.