Has there ever been an exorcism conducted by the Mormon Church?

They are officially recognized, yes. That being said, the record of them is seldom recorded to be re-shared. These principalities and powers of evil are best left out of official records and not shared publicly.

Know this, though, the Priesthood has the power to literally move mountains.

As a missionary in Chicago in the early 2000s, my companion and I were asked by an elderly member of the local congregation (called Wards) to give her granddaughter a blessing as there were no priesthood holders still alive in the family. This is something were were asked to do rather often, and so agreed with no problem. When we arrived we both felt that there was something dark and oppressive about the home, making us feel as though we were both being constantly watched, and we both felt like our skin was crawling. The young woman we were asked to bless refused to let us near her. She became violent, throwing anything she could get her hands on at us, and screaming obscenities at us. Her grandmother apologized to us and asked us to leave. We reported the incident to the Bishop of the Ward, and he made a visit of his own, only to meet a similar reception. Eventually an area authority became aware of the girl, and he determined that she was under possession of an evil spirit. Several leaders from the Ward, as well as the area authority, our Mission President, and several other missionaries, including myself and my companion were asked to come for an exorcism. We were told to join a circle around the girl while the Area Authority anointed her with olive oil and spoke a special blessing that we were told never to repeat to anyone outside that room. We were also told to never give specific details of what happened during the blessing so as not to empower the entity that we cast out further.

After the blessing, the girl improved and her life returned to normal. However, every one of the missionaries who participated, including myself, were either injured severely enough to be sent home from our missionary service, or became ill for such an extended period of time that were dismissed from service. During the months before I returned home after being clipped by a car while riding my bike down the street our apartment became a very uncomfortable place to live. We often heard strange noises outside our third story windows. Knocking on the outside walls of the building. Horrible smells from outside. Nothing ever actually happened inside of the apartment, but we always felt as though we were being watched from the windows.

So yes, Mormons do perform exorcisms. I participated in one, and afterward it seems that the entity involved did its best to make the lives of those who performed it rather miserable. A lot of people in this day and age feel that possession isn't real, and that it can all be explained by mental and medical disorders. I have to say, it's real. Nothing else can explain to me what happened that night. It's been fifteen years and I still have nightmares about it. I wouldn't want to talk about the details of what went on that night even if I hadn't been told not to.

There are many scriptures in the D&C that clarify this as a power that the restored priesthood of God has. Eg. D&C 84:67 In my name they shall cast out devils;

Here's an example of when Joseph Smith cast out a devil from Newel Knight: The Knights and the Trial of Joseph Smith.

It's not something that the church would publicize because it is a sacred subject. Having served as an LDS missionary, I can tell you missionaries have that same priesthood and can cast out devils. Technically, the LDS church believes it is the restored church, the same as when Jesus was on the earth, so all of those incidents would count as well.