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Context. The NASA exoplanet search mission Kepler is currently providing a wealth of light curves of ultra-high quality from space. Aims. We used high-quality Kepler photometry and spectroscopic data to investigate the Kepler target and binary candidate KIC 5988140. We aim to interpret the observed variations of KIC 5988140 considering three possible scenarios: binarity, co-existence of δ Sct- and γ Dor-type oscillations, and rotational modulation caused by an asymmetric surface intensity distribution. Methods. We used the spectrum synthesis method to derive the fundamental parameters Teff, log g, [M/H], and v sin i from the newly obtained high-resolution, high S/N spectra. Frequency analyses of both the photometric and the spectroscopic data were performed. Results. The star has a spectral type of A7.5 IV-III and a metallicity slightly lower than that of the Sun. Both Fourier analyses reveal the same two dominant frequencies F1 = 2F2 = 0.688 and F2 = 0.344 d-1. We also detected in the photometry the signal of nine more, significant frequencies located in the typical range of δ Sct pulsation. The light and radial velocity curves follow a similar, stable double-wave pattern which are not exactly in anti-phase but show a relative phase shift of about 0.1 period between the moment of minimum velocity and that of maximum light. Conclusions. Such findings are incompatible with the star being a binary system. We next show that, for all possible (limit) configurations of a spotted surface, the predicted light-to-velocity amplitude ratio is almost two orders larger than the observed value, which pleads against rotational modulation. The same argument also invalidates the explanation in terms of pulsations of type γ Dor (i.e. hybrid pulsations). We confirm the occurrence of various independent δ Sct-type pressure modes in the Kepler light curve. With respect to the low-frequency content, however, we argue that the physical cause of the remaining light and radial velocity variations of this late A-type star remains unexplained by any of the presently considered scenarios.