We use the Arecibo legacy fast ALFA (ALFALFA) 21cm survey to measure the number density of galaxies as a function of their rotational velocity, $V_\mathrm{rot,HI}$ (as inferred from the width of their 21cm emission line). Based on the measured velocity function we statistically connect galaxies with their host halo, via abundance matching. In a lambda cold dark matter ($��$CDM) cosmology, dwarf galaxies are expected to be hosted by halos that are significantly more massive than indicated by the measured galactic velocity; if smaller halos were allowed to host galaxies, then ALFALFA would measure a much higher galactic number density. We then seek observational verification of this predicted trend by analyzing the kinematics of a literature sample of gas-rich dwarf galaxies. We find that galaxies with $V_\mathrm{rot,HI} \lesssim 25$ $\mathrm{km} \, \mathrm{s}^{-1}$ are kinematically incompatible with their predicted $��$CDM host halos, in the sense that hosts are too massive to be accommodated within the measured galactic rotation curves. This issue is analogous to the "too big to fail" problem faced by the bright satellites of the Milky Way, but here it concerns extreme dwarf galaxies in the field. Consequently, solutions based on satellite-specific processes are not applicable in this context. Our result confirms the findings of previous studies based on optical survey data and addresses a number of observational systematics present in these works. Furthermore, we point out the assumptions and uncertainties that could strongly affect our conclusions. We show that the two most important among them -namely baryonic effects on the abundances of halos and on the rotation curves of halos- do not seem capable of resolving the reported discrepancy.

v3 matches the version published in A&A. Main differences with v2 are in Secs 3.2 & 4.4 and the addition of Appendix B. 11 figures, 14 pages (+2 appendices)