Yes, though it is very rarely suggested.
Women are more likely to develop Coccydynia due to its association with childbirth. However, it is a rare type of pain, which usually goes away with time. Coccyx plays an important role in supporting our weight when we sit. In medical literature, there are various suggestions regarding the importance of non-surgical treatment that should be tried before recommending surgery.
If the pain is persistent and is not alleviated with the treatments and/or activity changes, surgical removal of the coccyx, called Coccygectomy, is considered as an option. Yet, it is still rarely recommended and performed. You may either remove the complete coccyx or just a part of it.
Reviews of the successful operations suggest that there were no disadvantages to the patient caused by the absence of coccyx. However, the most common complication is the infection that spreads post-operation. While the surgery is relatively a simple procedure, the recovery can pose great difficulty for the patient as it is a long and uncomfortable process.