Can Space Be Infinitely Divided?

How many times can I half the distance between my hands? Assume perfect coordination and the ability to localize my palms to the quantum level. 15 halvings gets them to within a cell’s width. 33 to within a single atom, 50 and they’re a proton’s width apart. Half the distance 115 times and they’re a single Planck-length apart - 1.6x10^-35 meters. Surely we can keep going - .8, .4, .2 x10^-35 m? Bizarrely, those distances might not even exist in any meaningful way.