The save is made once per movement, per application of ball bearings.
The description of ball bearings says:
A creature moving across the covered area must succeed on a DC 10 Dexterity saving throw or fall prone.
"A creature moving across the covered area" refers to the event of a creature moving across the area - the entire event. Passing the saving throw means that the rest of my movement across the area is unimpeded.
If I stop in the middle of the area, then start my movement again, this is an entirely new instance of "a creature moving across the covered area", and I must make the saving throw again.
Now, if there is a second patch of ball bearings immediately adjacent to the first, then I'd say it requires a new saving throw. It is a second instance of the item's feature coming into effect, so it must be obeyed a second time. The idea here is that one dex save is good enough to navigate one application of ball bearings. Navigating twice as many ball bearings takes twice as much dexterity, or at least, twice as many attempts to not fall over.