Soft mechanical metamaterials can support a rich set of dynamic responses, which, to date, have received relatively little attention. Here, we report experimental, numerical, and analytical results describing the behavior of an anisotropic two-dimensional flexible mechanical metamaterial when subjected to impact loading. We not only observe the propagation of elastic vector solitons with three components—two translational and one rotational—that are coupled together, but also very rich direction-dependent behaviors such as the formation of sound bullets and the separation of pulses into different solitary modes.