Can one rise through a free-falling rope?

Yes, it is possible to rise theoretically w.r.t a ground frame. But the rope-man system's center of mass must keep moving downwards because of the only external force acting on the system (gravity). The lighter $M$ gets, the harder it will be for the man to rise, and it will become impossible in the limit the rope becomes mass-less. This is intuitive because then the center of mass of the system is effectively that of the man's alone, and it must therefore move down only.