The ability to remotely control a free-floating object through surface flows on a fluid medium can facilitate numerous applications. Current studies on this problem have been limited to uni-directional motion control due to the challenging nature of the control problem. Analytical modelling of the object dynamics is difficult due to the high-dimensionality and mixing of the surface flows while the control problem is hard due to the nonlinear slow dynamics of the fluid medium, underactuation, and chaotic regions. This study presents a methodology for manipulation of free-floating objects using large-scale physical experimentation and recent advances in deep reinforcement learning. We demonstrate our methodology through the open-loop control of a free-floating object in water using a robotic arm. Our learned control policy is relatively quick to obtain, highly data efficient, and easily scalable to a higher-dimensional parameter space and/or experimental scenarios. Our results show the potential of data-driven approaches for solving and analyzing highly complex nonlinear control problems.