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The action of four-dimensional Horndeski gravity coupled to Born-Infeld electromagnetic fields is given via the Kaluza-Klein process. Dyonic black hole solution of the theory is constructed. The metric is devoid of singularity at the origin independent of the parameter selections, this property is different from the one of Einstein-Born-Infeld black holes. Thermodynamics of the black hole is studied, thermodynamic quantities are calculated and the first law is checked to be satisfied. Thermodynamic phase transitions of the black holes are studied in extended phase space.