Can you kill the Drake in the Undead Burg?

Yes you can indeed kill him, and you get 10 000 souls when you do. As far as being killed with arrows, unfortunately I could not get a clear answer. The consensus on the forums I've looked through seems to be that he'll get down to about half health, at which point he'll cover his head, seemingly healing himself. There have also been several glitches reported where he died randomly.

Here is a video of somebody accomplishing the task:

You can kill the dragon. In fact, I managed to do it with a single arrow, and I was pretty low level. I'm not sure if it falls under one of the glitches mentioned in the other answer though.

Here's how I did it:

After going through the short section under the bridge and climbing the ladder past the rats, there's a staircase immediately on your left that goes up a tower. At the top of a tower is a Black Knight. If you can deal with the Knight, you can get a clear shot at the dragon from here. I'm not sure if the dragon was sleeping or what, but after I shot my first arrow, it looked the the dragon got startled and jumped of the bridge he was on. I thought he was just circling around or something, but a few seconds later, I had collected the 10,000 souls and the dragon was no more.

Also, you can totally kill the two dragons in Demon's Souls. Just not in 1-1, they will run away there. The Red Dragon can be killed in 1-2 and the Blue Dragon can be killed in 1-4.

We killed the Drake tonight through a combination of:

Shoot Drake with an arrow, wait a moment to be sure he comes down to bridge. Dash out onto bridge and strike his head, then quickly run back down the stairs. Sometimes we could get two strikes in before we ran back down the stairs to safety. Repeat process as needed until he is defeated. Be careful, if you step onto bridge too soon after shooting him he will spit fire from the top of tower. Sword used: Drakes tail, dealt about 111 points per hit. Arrows: he just used standard arrows, they dealt 5 points damage per hit. To get the DrakeSword go to the lower area of the bridge and shoot the Drake's tail anywhere between 20 to 50 times, depending on the arrows used it could be more, after enough hits the DrakeSword will automatically appear in your inventory.