Context. As is the case for all techniques involved in the research for exoplanets, direct imaging has to take into account the probability of so-called astrophysical false positives, which are phenomena that mimic the signature of the objects we are seeking. Aims. In this work we present the case of a false positive found during a direct-imaging survey conducted with VLT/NACO. A promising exoplanet candidate was detected around the K2-type star HD 8049 in July 2010. Its contrast of ΔH = 7.05 at 1.57 arcsec allowed us to assume a 35 MJup companion at 50 projected AU, for the nominal system age and heliocentric distance. Methods. To check whether it was gravitationally bound to the host star, as opposed to an unrelated background object, we re-observed the system one year later and concluded a high probability of a bound system. We also used radial velocity measurements of the host star, spanning a time range of ~30 yr, to constrain the companion's mass and orbital properties, as well as to probe the host star's spectral age indicators and general spectral energy distribution. We also obtained U-band imaging with EFOSC and near-infrared spectroscopy for the companion. Results. Combining all these information we conclude that the companion of HD 8049 is a white dwarf (WD) with temperature Teff = 18 800 ± 2100 K and mass MWD = 0.56 ± 0.08 M. The significant radial velocity trend combined with the imaging data indicates that the most probable orbit has a semi-major axis of about 50 AU. The discrepancy between the age indicators speaks against a bona-fide young star. The moderately high level of chromospheric activity and fast rotation, mimicking the properties of a young star, might be induced by the exchange of mass with the progenitor of the WD. This example demonstrates some of the challenges in determining accurate age estimates and identifications of faint companions.