|You say your employee does her job well, but the truth is that she only does part of it well. She is a good technician, and if her job description required no human contact, then you could say she is doing her job well. |
The problem is that you are dealing with personality issues rather than technical skills. When skills fall short, you can typically find a training program and be on your way. When you are dealing with personality issues like arrogance and condescension, you need more than a training program. However, that should not stop you. After all, personality issues are a major source of failure on the job.
Although this is a trying situation, you need to do more than try to give this employee some feedback. Whenever you see her flashing her arrogance or similar behaviors, give her specific feedback on the spot, followed by an example of a better way to act. You can raise the stakes by documenting her behaviors and letting her know the consequences if they continue.
You may also be able to find some programs or professional services that could help her. She is obviously developing enemies at work, and she needs to understand that she is the biggest one.