  A small fraction of GRBs with available data down to soft X-rays ($\sim0.5$
keV) have been shown to feature a spectral break in the low-energy part
($\sim1-10$ keV) of their prompt emission spectrum. The overall spectral shape
is consistent with optically thin synchrotron emission from a population of
particles that have cooled on a timescale comparable to the dynamic time to
energies that are still much higher than their rest mass energy (marginally
fast cooling regime). We consider a hadronic scenario and investigate if the
prompt emission of these GRBs can originate from relativistic protons that
radiate synchrotron in the marginally fast cooling regime. Using
semi-analytical methods, we derive the source parameters, such as magnetic
field strength and proton luminosity, and calculate the high-energy neutrino
emission expected in this scenario. We also investigate how the emission of
secondary pairs produced by photopion interactions and $\gamma\gamma$ pair
production affect the broadband photon spectrum. We support our findings with
detailed numerical calculations. Strong modification of the photon spectrum
below the break energy due to the synchrotron emission of secondary pairs is
found, unless the bulk Lorentz factor is very large ($\Gamma \gtrsim 10^3$).
Moreover, this scenario predicts unreasonably high Poynting luminosities
because of the strong magnetic fields ($10^6-10^7$ G) that are necessary for
the incomplete proton cooling. Our results strongly disfavor marginally fast
cooling protons as an explanation of the low-energy spectral break in the
prompt GRB spectra.
