You have to use a special black paint designed for ceilings so that when it falls to the ground it can just be swept up.

https://www.grandmashousediy.com/basement-ceilings-black/

You can still use a drop ceiling, only take maybe two inches, and simply frame out a bulkhead for around where the pipes drop below the ceiling height.

My son painted his ceiling in his basement family room a soft black. Not a harsh black. He sprayed it. Paint on the concrete was of no concern because he put in flooring. The soft black really does make all the duct work, wiring, etc all blend together. The ceiling recedes and looks higher. I was surprised at how well it looks.

IMO black ceilings in a basement gobble light and I think exposed pipes only work in lofts with 20’ ceilings . Dropped ceilings are now pretty attractive and for sure nicer than a bunch of exposed not very attractive pipes . Check out the options for ceilings there are many and IMO finish all the ceiling in the basement not just one space .

You could install a gypsum board (drywall or sheetrock) ceiling to the underside of the existing floor joist and box around the duct work creating a drop soffit. This would keep your ceiling height throughout most of the room. Would suggest painting the ceiling white and not black especially if there are no windows. Good luck!