  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
($\Lambda$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 $\Lambda$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.
