The magnetic properties of a nanonetwork consisting of ultrathin Ni nanowires (diameter < 4 nm) and Ni nanoballs (diameter < 20 nm) are studied at different stages of its growth during laser ablation in a superfluid helium medium. It has been established that, at the early stages of ablation, the nanonetwork consists mainly of nanowires and has a rectangular magnetic hysteresis loop. At the late stages of ablation, the concentration of nanoballs and their diameter increase, and the shape of the hysteresis loop deviates from a rectangular one. The fractal dimension of the nanonetwork is determined, which varies from 1 in the early stages of ablation, when individual nanowires occur, to 2, when the nanonetwork becomes so dense that it is a continuous film. It is shown that the saturation magnetization changes with a change in the fractality of the nanonetwork, which, under constant ablation conditions, is explained by the transformation of nanowires into nanoballs during their folding.