When a force deforms an elastic object, practical experience suggests that the resulting displacement will be in the same direction as the force. This property is known as positive stiffness. Less familiar is the concept of negative stiffness, where the deforming force and the resulting displacement are in opposite directions. (Negative stiffness is distinct from negative Poisson's ratio, which refers to the occurrence of lateral expansion upon stretching an object.) Negative stiffness can occur, for example, when the deforming object has stored (or is supplied with) energy. This property is usually unstable, but it has been shown theoretically that inclusions of negative stiffness can be stabilized within a positive-stiffness matrix. Here we describe the experimental realization of this composite approach by embedding negative-stiffness inclusions of ferroelastic vanadium dioxide in a pure tin matrix. The resulting composites exhibit extreme mechanical damping and large anomalies in stiffness, as a consequence of the high local strains that result from the inclusions deforming more than the composite as a whole. Moreover, for certain temperature ranges, the negative-stiffness inclusions are more effective than diamond inclusions for increasing the overall composite stiffness. We expect that such composites could be useful as high damping materials, as stiff structural elements or for actuator-type applications.