A recent paper in Physical Review Letters shows that the answer to this question is yes! The authors observe repeated bouncing of an oil droplet within a stratified ethanol-water mixture, which continues over many minutes. I will describe their experimental set-up, their proposed mechanism (which involves Marangoni stress at the droplet-liquid interface), and the results of their numerical modelling of the process.
Li, Diddens, Prosperetti, Chong, Zhang, Lohse. "Bouncing Oil Droplet in a Stratified Liquid and its Sudden Death." Physical Review Letters 122: 154502 (2019)

Link to paper