  The observed hard TeV gamma-ray spectrum of the nearby gamma-ray burst (GRB)
190829A may challenge the conventional leptonic GRB afterglow model. It has
been proposed that an ultra-high-energy (UHE; $\varepsilon^{'}_{\rm p}\sim
10^{20}$ eV) proton population can be pre-accelerated by internal shocks in GRB
jets. We study possible signatures of the UHE protons embedded in the TeV
afterglows when they escape the afterglow fireball. We show that the leptonic
model can represent the observed multiwavelength lightcurves and spectral
energy distributions of GRB 190829A by considering the uncertainties of the
model parameters. Attributing the TeV gamma-ray afterglows to the emission of
both the electron self-Compton scattering process and the UHE proton
synchrotron radiations in the afterglow fireball, we obtain tentative upper
limits of $\log_{10} \varepsilon_{\rm p}^{\prime}/{\rm eV}\sim 20.46$ and
$\log_{10}E_{\rm p, total}/{\rm erg}\leq 50.75$, where $E_{\rm p, total}$ is
the total energy of the proton population. The synchrotron radiations of the
UHE protons should dominate the early TeV gamma-ray afterglows, implying that
early observations are critical for revealing the UHE proton population.
