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The radiative energy loss of fast partons traveling through the quark-gluon plasma (QGP) is commonly studied within perturbative QCD (pQCD). Nonperturbative (NP) effects, which are expected to become important near the critical temperature, have been much less investigated. Here, we utilize a recently developed $T$-matrix approach to incorporate NP effects for gluon emission off heavy quarks propagating through the QGP. We set up four cases that contain, starting from a Born diagram calculation with color-Coulomb interaction, an increasing level of NP components, by subsequently including (remnants of) confining interactions, resummation in the heavy-light scattering amplitude, and off-shell spectral functions for both heavy and light partons. For each case we compute the power spectra of the emitted gluons, heavy-quark transport coefficients (drag and transverse-momentum broadening, $\hat{q}$), and the path-length dependent energy loss within a "QGP brick" at fixed temperature. Investigating the differences in these quantities between the four cases illustrates how NP mechanisms affect gluon radiation processes. While the baseline perturbative processes experience a strong suppression of soft radiation due to thermal masses of the emitted gluons, confining interactions, ladder resummations and broad spectral functions (re-)generate a large enhancement toward low momenta and low temperatures. For example, for a 10 GeV charm quark at 200 MeV temperature, they enhance the transport coefficients by up to a factor of 10, while the results smoothly converge to perturbative results at sufficiently hard scales.