No, a K-1 visa cannot be extended. The K-1 visa is a single-entry visa that allows the non-citizen fiancé(e) to enter the United States to marry their U.S. citizen partner within 90 days of arrival. Once the 90-day period is over, the K-1 visa automatically expires.
If the couple does not get married within the 90-day timeframe, the non-citizen fiancé(e) must leave the United States. Failure to depart the U.S. in this situation may result in future immigration problems, including denial of adjustment of status and possible removal proceedings.
If the couple cannot marry within 90 days due to unforeseen circumstances, they may need to restart the K-1 visa application process or explore other visa options, such as a spousal visa (CR1 or IR1), depending on their situation.