(Abridged) Aims: In this paper, we seek to establish the suitability of imaging spectroscopy performed in the Ca II 854.2 nm line as a means to investigate the solar chromosphere at high resolution. Methods: We utilize monochromatic images obtained with the Interferometric BIdimensional Spectrometer (IBIS) at multiple wavelengths within the Ca II 854.2 nm line and over several quiet areas. We analyze both the morphological properties derived from narrow-band monochromatic images and the average spectral properties of distinct solar features such as network points, internetwork areas and fibrils. Results: The spectral properties derived over quiet-Sun targets are in full agreement with earlier results obtained with fixed-slit spectrographic observations, highlighting the reliability of the spectral information obtained with IBIS. Furthermore, the very narrowband IBIS imaging reveals with much clarity the dual nature of the Ca II 854.2 nm line: its outer wings gradually sample the solar photosphere, while the core is a purely chromospheric indicator. The latter displays a wealth of fine structures including bright points, akin to the Ca II H2V and K2V grains, as well as fibrils originating from even the smallest magnetic elements. The fibrils occupy a large fraction of the observed field of view even in the quiet regions, and clearly outline atmospheric volumes with different dynamical properties, strongly dependent on the local magnetic topology. This highlights the fact that 1-D models stratified along the vertical direction can provide only a very limited representation of the actual chromospheric physics.