In this work, we propose a new non-Abelian generalization of the Born-Infeld Lagrangian. It is based on a geometrical property of the Abelian Born-Infeld Lagrangian in its determinantal form. Our goal is to extend the Abelian second type Born-Infeld action to the non-Abelian form preserving this geometrical property, which permits us to compute the generalized volume element as a linear combination of the components of metric and the Yang-Mills energy-momentum tensors. Under the BPS-like condition, the action proposed reduces to that of the Yang-Mills theory, independently of the gauge group. New instanton-wormhole solution and static and spherically symmetric solution in curved spacetime for an $SU(2)$ isotopic ansatz are solved and the $N = 1$ supersymmetric extension of the model is performed. The uniqueness of the theory from the physical and geometrical point of view is discussed in the light of new results obtained.