Extremely metal-poor (XMP) galaxies are chemically, and possibly dynamically, primordial objects in the local Universe. Our objective is to characterize the HI content of the XMP galaxies as a class, using as a reference the list of 140 known local XMPs compiled by Morales-Luis et al. (2011). We have observed 29 XMPs, which had not been observed before at 21 cm, using the Effelsberg radio telescope. This information was complemented with HI data published in literature for a further 53 XMPs. In addition, optical data from the literature provided morphologies, stellar masses, star-formation rates and metallicities. Effelsberg HI integrated flux densities are between 1 and 15 Jy km/s, while line widths are between 20 and 120 km/s. HI integrated flux densities and line widths from literature are in the range 0.1 - 200 Jy km/s and 15 - 150 km/s, respectively. Of the 10 new Effelsberg detections, two sources show an asymmetric double-horn profile, while the remaining sources show either asymmetric (7 sources) or symmetric (1 source) single-peak 21 cm line profiles. An asymmetry in the HI line profile is systematically accompanied by an asymmetry in the optical morphology. Typically, the g-band stellar mass-to-light ratios are ~0.1, whereas the HI gas mass-to-light ratios may be up to 2 orders of magnitude larger. Moreover, HI gas-to-stellar mass ratios fall typically between 10 and 20, denoting that XMPs are extremely gas-rich. We find an anti-correlation between the HI gas mass-to-light ratio and the luminosity, whereby fainter XMPs are more gas-rich than brighter XMPs, suggesting that brighter sources have converted a larger fraction of their HI gas into stars. The dynamical masses inferred from the HI line widths imply that the stellar mass does not exceed 5% of the dynamical mass, while the ion{H}{i} mass constitutes between 20 and 60% of the dynamical mass. (abridged)
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