Context. For over 12 yr, we have carried out a precise radial velocity (RV) survey of a sample of 373 G- and K-giant stars using the Hamilton Échelle Spectrograph at the Lick Observatory. There are, among others, a number of multiple planetary systems in our sample as well as several planetary candidates in stellar binaries. Aims: We aim at detecting and characterizing substellar and stellar companions to the giant star HD 59686 A (HR 2877, HIP 36616). Methods: We obtained high-precision RV measurements of the star HD 59686 A. By fitting a Keplerian model to the periodic changes in the RVs, we can assess the nature of companions in the system. To distinguish between RV variations that are due to non-radial pulsation or stellar spots, we used infrared RVs taken with the CRIRES spectrograph at the Very Large Telescope. Additionally, to characterize the system in more detail, we obtained high-resolution images with LMIRCam at the Large Binocular Telescope. Results: We report the probable discovery of a giant planet with a mass of mp sin i = 6.92-0.24+0.18 MJup orbiting at ap = 1.0860-0.0007+0.0006 au from the giant star HD 59686 A. In addition to the planetary signal, we discovered an eccentric (eB = 0.729-0.003+0.004) binary companion with a mass of mB sin i = 0.5296-0.0008+0.0011 M⊙ orbiting at a close separation from the giant primary with a semi-major axis of aB = 13.56-0.14+0.18 au. Conclusions: The existence of the planet HD 59686 Ab in a tight eccentric binary system severely challenges standard giant planet formation theories and requires substantial improvements to such theories in tight binaries. Otherwise, alternative planet formation scenarios such as second-generation planets or dynamical interactions in an early phase of the system's lifetime need to be seriously considered to better understand the origin of this enigmatic planet. Based on observations collected at the Lick Observatory, University of California.Based on observations collected at the Large Binocular Telescope, on Mount Graham, Arizona.RV data (Table A.1) are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/595/A55