We demonstrate experimentally, numerically, and analytically that soft architected materials can support the propagation of elastic vector solitons. More specifically, we focus on structures comprising a network of squares connected by thin and highly deformable ligaments and investigate the propagation of planar nonlinear elastic waves. We find that for sufficiently large amplitudes two components—one translational and one rotational—are coupled together and copropagate without dispersion. Our results not only show that soft architected materials offer a new and rich platform to study the propagation of nonlinear waves, but also open avenues for the design of a new generation of smart systems that take advantage of nonlinearities to control and manipulate the propagation of large amplitude vibrations.