The dynamics of extended many-body systems are generically chaotic. Classically, a hallmark of chaos is the exponential sensitivity to initial conditions captured by positive Lyapunov exponents. Supplementing chaotic dynamics with stochastic resetting drives a sharp dynamical phase transition: we show that the Lyapunov spectrum, i.e., the complete set of Lyapunov exponents, abruptly collapses to zero above a critical resetting rate. At criticality, we find a sudden loss of analyticity of the velocity-dependent Lyapunov exponent, which we relate to the transition from ballistic scrambling of information to an arrested regime where information becomes exponentially localized over a characteristic length diverging at criticality with an exponent ν=1/2 and a dynamical exponent z=2. We illustrate our analytical results on generic chaotic dynamics by numerical simulations of coupled map lattices.