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 $gammagamma$ 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 disfavour marginally fast cooling protons as an explanation of the low-energy spectral break in the prompt GRB spectra.
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