We present ultra-high-resolution (0.35 km s−1 FWHM) observations of the interstellar Ca K line towards seven nearby stars. The spectral resolution was sufficient to resolve the line profiles fully, thereby enabling us to detect hitherto unresolved velocity components, and to obtain accurate measurements of the velocity dispersions (b values). Absorption components with velocities similar to those expected for the Local Interstellar Cloud (LIC) and the closely associated ‘G cloud’ were identified towards six of the seven stars. However, in most cases the b values deduced for these components were significantly larger than the b ≈ 2.2 km s−1 (i.e. Tk ≈ 7000 K, vt ≈ 1 km s−1) expected for the LIC, and it is argued that this results from the presence of additional, spectrally unresolved, components having similar velocities and physical conditions. For two stars (δ Vel and α Pav) we detect interstellar components with much smaller b values (1.1 ± 0.3 and 0.8 ± 0.1 km s−1, respectively) than are expected for low-density clouds within the Local Bubble. In the case of the narrow α Pav component, we also find an anomalously large Na i/Ca ii column density ratio, which is indicative of a relatively high density. Thus it is possible that, in addition to LIC-type clouds, the local interstellar medium contains a population of previously undetected cooler and denser interstellar clouds
