We have studied the growth dynamics of ultrasmooth hydrogenated amorphous carbon films deposited on silicon substrates by electron cyclotron resonance chemical vapor deposition from argon/methane gas mixtures applying a high negative external bias. The surface morphology of films deposited for different growth times under the same experimental conditions was analyzed by atomic force microscopy. Our analysis leads to values of the growth, roughness, and coarsening exponents of 0, 0.1, and 0.5, respectively. As it has been recently proposed that the growth dynamics of amorphous films by ion-assisted methods should obey the Edwards-Wilkinson (EW) growth mode, we have analyzed the compatibility of our data with this model. Our analysis indicates that, although the scaling data could be interpreted in terms of the EW model, the relative large data error bars and the film ultrasmoothness preclude the unambiguous assessment of the EW growth mode for our film growth evolution. In our system, the interplay of shadowing, physical sputtering and enhanced surface mobility ion-induced effects contribute likely to the leveling and final ultrasmoothness of the film surface.