We investigate a non-local extension of Yang–Mills theory in four-dimensional Euclidean spacetime based on an entire-function form factor acting on the gauge kinetic operator. The modification introduces exponential suppression at high momentum scales while preserving gauge invariance and avoiding additional propagator poles. The resulting framework provides a natural ultraviolet regularization and suggests the dynamical emergence of a mass gap in the gauge sector. Using functional integral methods, spectral analysis, and a Gribov-type infrared modification, we derive a closed expression for the effective mass scale controlling the lowest excitation of the theory. The resulting scale lies in the range 1–2 GeV for physically reasonable parameter choices, consistent with lattice estimates for the scalar glueball mass. This framework suggests that non-local entire-function regularizations may provide a mathematically consistent path toward understanding mass gap generation in non-Abelian gauge theories.