What a remarkable life has been lived, by anyone who can say "No!" to this question, @galenkp:

"Have you ever had to deal with a difficult person?"

If they exist, not being one of those remarkable people myself, here is a quick bullet list of what I have found to be helpful:

• Stick to business. Keep it objective. Don't respond to anything personal. Particularly if they are attempting to "push your buttons," don't rise to the bait.
• Find areas of agreement. Focus on what unites you. Rather than divides you. Establishing common ground reduces tension.
• Avoid the zero-sum game of an all or nothing approach. Give credit where credit is due, e.g. "yes, that is a good point." Be genuine and sincere in this. It will set you apart.
• Smile. Speak in calm, quiet, professional tones, no matter how they are talking to you.

While these doesn't eliminate the difficulties, it will almost certainly reduce them. Hopefully to a reasonably manageable level.