You have a dull drill bit. A sharp bit will chew its way through copper with no problems. For the perfectly drilled hole, follow these directions:
1) Make a tiny dent with either an automatic center punch found at any tool supplier, or even a nail and hammer. This creates a guide for the drill bit so it won't skip over the metal.
2) Lube your bit with Bur Life (waxy stick) or better the liquid variety, Bur Lube. A quick dip will do you fine. The feature of the liquid type is that not only do you create lubrication for your bit, but you maintain the sharp edges, causing the drill bit to "rip" through the metal. Just dab of the excess or spin it off in another container.
3) Once you have drilled through your metal, every other time you drill, dunk your drill bit in the lubricant and pulse the flexshaft just a little. The centrifugal action will spin off any loose metal bits and they will fall to the bottom of your container. Grit or metal shards will dull your drill bit quickly. The stick variety of lubricant is good for sawblades, but NOT drill bits or burs. If you plunge a clogged bur into the hardened lubricant, you are just transferring more grit back to the source. You need it OFF your drill bit or bur so it can continue staying sharp and doing its job.
4) Drill your hole. You will see a marked difference in a dull bit and a sharp one.
5) What are the best drill bits? Preferably, I like HSS which stands for High Speed Steel which are sharp and last longer, provide you take care of them. They are a little more expensive, but they last longer under higher speeds and friction.
Lastly, when you are at the "dent", make sure your drill bit is moving. You will ensure that the guide will do its job and you need a smaller dent. Keep a light hand on the handpiece and then lightly push. Let the drill bit do its job.