Keep the part COOL! Copper readily conducts heat and as you drill deeper, the part expands (from heat) and pinches the drill body and causes bad things to happen as you describe. I'd suggest:
1) Flood coolant of both the drill and the part.
2) Decrease feed-rate.
3) Peck-drill to break the chips and allow better coolant penetration.
4) Worst case workaround is to relieve the lands of the drill to reduce friction.
Hope this helps.