Damping mechanical resonances is a formidable challenge in an increasing number of applications. Many passive damping methods rely on using low stiffness, complex mechanical structures or electrical systems, which render them unfeasible in many of these applications. Herein, a new method for passive vibration damping, by allowing buckling of the primary load path in mechanical metamaterials and lattice structures, is introduced, which sets an upper limit for vibration transmission: the transmitted acceleration saturates at a maximum value in both tension and compression, no matter what the input acceleration is. This nonlinear mechanism leads to an extreme damping coefficient tanδ ≈ 0.23 in a metal metamaterial—orders of magnitude larger than the linear damping coefficient of traditional lightweight structural materials. This principle is demonstrated experimentally and numerically in free‐standing rubber and metal mechanical metamaterials over a range of accelerations. It is also shown that damping nonlinearities even allow buckling‐based vibration damping to work in tension, and that bidirectional buckling can further improve its performance. Buckling metamaterials pave the way toward extreme vibration damping without mass or stiffness penalty, and, as such, could be applicable in a multitude of high‐tech applications, including aerospace, vehicles, and sensitive instruments.