We report measurements of the resistivity in the ferromagnetic state of epitaxial thin films of La(1-x)Ca(x)MnO3 and the low-temperature specific heat of a polycrystalline La0.8Ca0.2MnO3. The resistivity below 100 K can be well fitted by rho-rho(0) = Eomega(s)/sinh (2)(hs/ 2k(B)T) with h(omega)(s)/k(B) approximately 80 K and E being a constant. Such behavior is consistent with small-polaron coherent motion which involves a relaxation due to a soft optical phonon mode that is strongly coupled to the carriers. The specific-heat data also suggest the existence of such a phonon mode. The present results thus provide evidence for small-polaron metallic conduction in the ferromagnetic state of manganites.