We estimate the cosmic number density of the recently identified class of HI-bearing ultra-diffuse sources (HUDs) based on the completeness limits of the ALFALFA survey. These objects have HI masses approximately in the range 8.5 < logM_HI∕M_⊙ < 9.5, average r-band surface brightnesses fainter than 24 mag arcsec⁻², half-light radii greater than 1.5 kpc, and are separated from neighbours by at least 350 kpc. In this work we demonstrate that they contribute at most ~6% of the population of HI-bearing dwarfs detected by ALFALFA (with similar HI masses), have a total cosmic number density of (1.5 ± 0.6) × 10⁻³ Mpc⁻³, and an HI mass density of (6.0 ± 0.8) × 10⁵ M_⊙ Mpc⁻³. We estimate that this is similar to the total cosmic number density of ultra-diffuse galaxies (UDGs) in groups and clusters, and conclude that the relation between the number of UDGs hosted in a halo and the halo mass must have a break below M₂₀₀ ~ 10¹² M_⊙ in order to account for the abundance of HUDs in the field. The distribution of the velocity widths of HUDs rises steeply towards low values, indicating a preference for slow rotation rates compared to the global HI-rich dwarf population. These objects were already included in previous measurements of the HI mass function, but have been absent from measurements of the galaxy stellar mass function owing to their low surface brightness. However, we estimate that due to their low number density the inclusion of HUDs would constitute a correction of less than 1%. Comparison with the Santa Cruz semi-analytic model shows that it produces HI-rich central UDGs that have similar colours to HUDs, but that these UDGs are currently produced in a much greater number. While previous results from this sample have favoured formation scenarios where HUDs form in high spin-parameter halos, comparisons with recent results which invoke that formation mechanism reveal that this model produces an order of magnitude more field UDGs than we observe in the HUD population, and these have an occurrence rate (relative to other dwarfs) that is approximately double what we observe. In addition, the colours of HUDs are bluer than predicted, although we suspect this is due to a systematic problem in reproducing the star formation histories of low-mass galaxies rather than being specific to the ultra-diffuse nature of these sources.