Opportunistic mobile ad hoc networks (MANETs) are a special class of sparse and disconnected MANETs where data communication exploits sporadic contact opportunities among nodes. We consider opportunistic MANETs where nodes move independently at random over a square of the plane. Nodes exchange data if they are at a distance at most r within each other, where r > 0 is the node transmission radius. The flooding time is the number of time-steps required to broadcast a message from a source node to every node of the network. Flooding time is an important measure of how fast information can spread in dynamic networks. We derive the first upper bound on the flooding time, which is a decreasing function of the maximal speed of the nodes. The bound holds with high probability, and it is nearly tight. Our bound shows that, thanks to node mobility, even when the network is sparse and disconnected, information spreading can be fast.