I'm guessing that we are assuming paint particles are infinitesimal and don't clump together
as with all things infinity leave common sense at the door.
talking about litterally infinitesimal particles leads to madness,. example: how many such paint particles would it take to fill a 10 litre bucket?
An infinite amount.
no matter if you kept pouring more and more into your bucket it would remain almost totally empty.
It doesn't even really make much sense to talk about volumes of these paint particles at all, how many are in an single cm^3, how about in a meter cubed, what's stopping however many are in a meter cubed from being squished down to a cm cubed?
Forget anything about painting an infinite surface area with these particles, filling a finite recular bucket would take an infinite amount of these paint particles.
So no real paradox, if you talk about infinitesimal particles of paint then talking about it's volume makes no sense.
If you're not talking about infinitesimal paint particles then the surface area they can reach and thus paint is finite.
Give a man a fish, he owes you one fish. Teach a man to fish, you give up your monopoly on fisheries.