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.