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Current gravitational-wave observations set the most stringent bounds on the abundance of primordial black holes (PBHs) in the solar mass range. This constraint, however, inherently relies on the merger rate predicted by PBH models. Previous analyses have focused mainly on two binary formation mechanisms: early Universe assembly out of decoupling from the Hubble expansion and dynamical capture in present-day dark matter structures. Using reaction rates of three-body processes studied in the astrophysical context, we show that, under conservative assumptions, three-body interactions in PBH halos efficiently produce binaries. Those binaries form at high redshift in Poisson-induced PBH small-scale structures and a fraction is predicted to coalesce and merge within the current age of the Universe, at odds with the dynamical capture scenario where they merge promptly. In general, we find that this channel predicts rates comparable to the dynamical capture scenario. However, binaries formed from three-body interactions do not significantly contribute to the overall PBH merger rate unless PBHs made up a dominant fraction of the dark matter above the solar mass range, a scenario that is ruled out by current constraints. Our results support strong bounds on the PBH abundance in the stellar mass range derived from Laser Interferometer Gravitational-Wave Observatory/Virgo/KAGRA observations. Finally, we show that both dynamical channels are always subdominant compared to early Universe assembly for PBH mergers in the asteroid mass range, while we expect it to become relevant in scenarios where PBHs are initially strongly clustered.