The celebrated Kardar-Parisi-Zhang (KPZ) equation describes the kinetic roughening of stochastically growing interfaces. In one dimension, the KPZ equation is exactly solvable and its statistical properties are known to an exquisite degree. Yet recent numerical simulations in the tensionless (or inviscid) limit of the KPZ equation [Phil. Trans. Roy. Soc. A 380 20210090 (2022), Phys. Rev. E 106 024802 (2022)] unveiled a new scaling, with a critical dynamical exponent z = 1 different from the KPZ one z = 3/2. In this Letter, we show that this scaling is controlled by a fixed point which had been missed so far and which corresponds to an infinite non-linear coupling. Using the functional renormalization group (FRG), we demonstrate the existence of this fixed point and show that it yields z = 1. We calculate the correlation function and associated scaling function at this fixed point, providing both a numerical solution of the FRG equations within a reliable approximation, and an exact asymptotic form obtained in the limit of large wavenumbers. Both scaling functions accurately match the one from the numerical simulations.