A modulated metamaterial that exhibits both time-periodic stiffness and mass simultaneously is presented. The metamaterial element includes a primary body that undergoes infinitesimal motion, and is connected to a dynamic-mechanism structure, involving a rotational body, and spring with a large-scale motion, which is designed to produce a time-modulated linear momentum and elastic constraint for the primary body. The non-reciprocal wave propagation is then investigated in a space–time lattice metamaterial that is constructed by coupling doubly time-modulated elements with linear springs of constant stiffness. The dispersion property shows the frequency degeneracy occurring at the center or edge of the Brillouin zone, and the unidirectional bandgap at certain frequencies. This phenomenon represents a unique property of the doubly modulated metamaterials compared to the singly modulated ones, thus may provide more promising applications to the design of non-reciprocal devices.