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Amorphous carbon (a-C) materials have diverse interesting and useful properties, but the understanding of their atomic-scale structures is still incomplete. Here, we report on extensive atomistic simulations of the deposition and growth of a-C films, describing interatomic interactions using a machine learning (ML) based Gaussian approximation potential model. We expand widely on our initial work [M. A. Caro et al., Phys. Rev. Lett. 120, 166101 (2018)] by now considering a broad range of incident ion energies, thus modeling samples that span the entire range from low-density (sp(2)-rich) to high-density (sp(3)-rich, "diamondlike") amorphous forms of carbon. Two different mechanisms are observed in these simulations, depending on the impact energy: low-energy impacts induce sp- and sp(2)-dominated growth directly around the impact site, whereas high-energy impacts induce peening. Furthermore, we propose and apply a scheme for computing the anisotropic elastic properties of the a-C films. Our work provides fundamental insight into this intriguing class of disordered solids, as well as a conceptual and methodological blueprint for simulating the atomic-scale deposition of other materials with ML driven molecular dynamics.