"When I look at this process in Service Management, it tells me it is running."
At what step is it running?
"I can also go into Workspace and see that it is running in there (I can also see the completed tasks and assignments). "
I'm not sure what you mean there.
"It seems to be the case that there are no outstanding tasks or assignments for the process, however the process itself is still running. Can this be the case?"
No. If there are no more step to do, the process shouldn't be running. You can also use the Record and Playback feature to monitor your process and see where it is. From Workbench you can click Record Process, then run your process until the end, and then play it to see what it did. This is a very useful feature.
Jasmin