Yes, there's a good chance you'll kill your raspberries. The runners are all connected to the parent plant.

You could just edge along the fence and cut all the runners free of the parents, then spray. That said, spray could hit your parent plants - since it is a spray. So take care to - maybe put a barrier along the fence.

Roundup is glyphosate. It translocates throughout the plant if it works as intended, which is likely to kill the mother plant. Glufosinate will kill the runner, but not translocate like glyposate. Some brands of glufosinate are Ignite, Finale, and Liberty. Ignite is often used in vineyards just for this reason, to kill runners without killing the mother vine. It is more effective on broadleaf weeds than grass and should be used when the weeds are small.