We consider the problem of extracting tripartite entanglement through single local instantaneous interactions of a separable target system A -B -C with a scalar field. We find, nonperturbatively, that tripartite entanglement is easily extracted in this scenario, in strong contrast to bipartite extraction, which is not possible due to a no-go theorem. Contrary to previous expectations, the tripartite entanglement is of the Greenberger-Horne-Zeilingertype and can be maximized by an optimal value of the coupling.