Context. Despite important advances in space asteroseismology during the last decade, the early phases of evolution of stars with masses above 15 M⊙ (including the O stars and their evolved descendants, the B supergiants) have been only vaguely explored up to now. This is due to the lack of adequate observations for a proper characterization of the complex spectroscopic and photometric variability occurring in these stars. Aim. Our goal is to detect, analyze, and interpret variability in the early-B-type supergiant HD 2905 (κ Cas, B1 Ia) using long-term, ground-based, high-resolution spectroscopy. Methods: We gather a total of 1141 high-resolution spectra covering some 2900 days with three different high-performance spectrographs attached to 1-2.6m telescopes at the Canary Islands observatories. We complement these observations with the hipparcos light curve, which includes 160 data points obtained during a time span of 1200 days. We investigate spectroscopic variability of up to 12 diagnostic lines by using the zero and first moments of the line profiles. We perform a frequency analysis of both the spectroscopic and photometric dataset using Scargle periodograms. We obtain single snapshot and time-dependent information about the stellar parameters and abundances by means of the FASTWIND stellar atmosphere code. Results: HD 2905 is a spectroscopic variable with peak-to-peak amplitudes in the zero and first moments of the photospheric lines of up to 15% and 30 km s-1, respectively. The amplitude of the line-profile variability is correlated with the line formation depth in the photosphere and wind. All investigated lines present complex temporal behavior indicative of multi-periodic variability with timescales of a few days to several weeks. No short-period (hourly) variations are detected. The Scargle periodograms of the hipparcos light curve and the first moment of purely photospheric lines reveal a low-frequency amplitude excess and a clear dominant frequency at 0.37 d-1. In the spectroscopy, several additional frequencies are present in the range 0.1-0.4 d-1. These may be associated with heat-driven gravity modes, convectively driven gravity waves, or sub-surface convective motions. Additional frequencies are detected below 0.1 d-1. In the particular case of Hα, these are produced by rotational modulation of a non-spherically symmetric stellar wind. Conclusions: Combined long-term uninterrupted space photometry with high-precision spectroscopy is the best strategy to unravel the complex low-frequency photospheric and wind variability of B supergiants. Three-dimensional (3D) simulations of waves and of convective motions in the sub-surface layers can shed light on a unique interpretation of the variability.