Check you have the right head in it puller (has it been swapped over by mistake), the rivet mandrel should be a snug fit in the gun.
If the inside diameter is too large when the rivet is pulled you can get part of the rivet pulled up between the mandrel and and the gun.
Telltale sign is a lip round the hole in the rivet where the mandrel pulls out.
I get this sometimes when I use the angle wedges on tight rivets as the hole is slightly larger than needed to go over the rivet mandrel.
Normally there should be no sharpness to the rivet at all.
The other option is a broken mandrel, is the mandrel breaking off proud of the surface and that is what you feel. This can be down to air pressure being used so you can try changing it to see if that helps.
Just noticed you say it is the second RV, could it just be worn out after the first one (the rivets are soft but the mandrel is steel so could have worn the head a bit).
It took me so long to reply I got beaten to it while writing the post lol.