We experimentally study the dynamics of weakly interacting Bose-Einstein condensates of cesium atoms in a 1D optical lattice with a periodic driving force. After a sudden start of the driving we observe the formation of stable wave packets at the center of the first Brillouin zone (BZ) in momentum space, and we interpret these as Floquet solitons in periodically driven systems. The wave packets become unstable when we add a trapping potential along the lattice direction leading to a redistribution of atoms within the BZ. The concept of a negative effective mass and the resulting changes to the interaction strength and effective trapping potential are used to explain the stability and the time evolution of the wave packets. We expect that similar states of matter waves exist for discrete breathers and other types of lattice solitons in periodically driven systems.