I cut mine with hand pruning shears. It's a tight fit but the blades slip in between the branch and the stem base that covers the top of the fruit. They just don't come off easily any other way.

Mine (Tam Kam) grow tight to the branch, I could never get a pruning shears in there. I use wire cutters.

If you try to pull them, the branches will break, as persimmon wood is brittle and/or the soft ones will squish in your hand. Yes, you have to cut them off.

Carla in Sac

with more persimmon trees than I need...I have no worries about cutting a small branch here and there.