Aims. We study the properties of low-column density gas clumps in the halo of the Milky Way based on high-resolution 21-cm observations.Methods. Using interferometric data from the Westerbork Synthesis Radio Telescope (WSRT) and the Very Large Array (VLA), we studied H i emission at low-, intermediate- and high radial velocities along four lines of sight towards the quasars QSO J00032323, QSO B1331+170, QSO B0450-1310, and J081331+254503. Along these sightlines we had previously detected weak Ca ii and Na i absorbers in the optical spectra of these quasars.Results. The analysis of the high-resolution H i data reveals several compact and cold clumps of neutral gas at velocities similar to the optical absorption. The clumps have narrow H i line widths in the range of 1.8 ≤ ≤ 13.0 km s^{-1^{, yielding upper limits for the kinetic temperature of the gas of 70 ≤ ≤ 3700 K. The neutral gas has low H i column densities in the range of 5 10^{18^{...3 10^{19^{cm^{-2^{. All clumps have angular sizes of only a few arcminutes.Conclusions. Our high-resolution 21-cm observations indicate that many of the Ca ii and Na i absorbers seen in our optical quasar spectra are associated with low-column density H i clumps on small angular scales. This suggests that, next to the massive, high-column density neutral gas clouds in the halo (the common 21-cm low-, intermediate-, and high-velocity clouds, LVCs, IVCs, and HVCs), a population of low-mass, neutral gas structures exists in the halo and remain mostly unseen in the existing 21-cm all-sky surveys of IVCs and HVCs. One of our absorbers may be associated with the Magellanic Stream, two intermediate-velocity clouds are probably part of the Intermediate-Velocity Spur and the Low-latitude IV arch, respectively. The remaining systems could be located either in the lower halo or in the disk of the Milky Way. The estimated thermal gas pressures of the detected H i clumps are consistent with what is expected from theoretical models of gas in the inner and outer Milky Way halos.}}}}}}}}

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