Foraging and other natural phenomena show superdiffusive transport that are described by Lévy walks. We present a model experimental system, an octanoic acid (OA) drop on aqueous unsaturated OA solutions, for studying Lévy walks in controlled environments. The drop spontaneously moves due to unbalanced Marangoni forces, in a manner that mimics biological propulsion in living systems. The drop motion is shown to be a Lévy walk by demonstrating superdiffusive transport, Lévy-stable distributed step lengths, and power-law decay in the velocity autocorrelation function, with consistent exponents. © 2020 American Physical Society.