The Basque beret is a traditional headwear associated with the Basque region of southwestern France and northern Spain. Its precise origin is not known with certainty, but it's likely that the beret evolved from wool caps worn by Basque shepherds for centuries.
The beret as we know it today appeared in the 19th century, and became popular among workers in the wool industry in France. It was also worn by French soldiers during World War I. The Basque beret has become a symbol of Basque culture, and is often associated with traditional dances and festivities.
It's true that shepherds from the Aspe and Ossau valleys began knitting wool berets for themselves and their neighbors, and that Basque sailors helped spread the beret throughout Spain and America.
Regarding the term "Basque beret", it's possible that Napoleon III helped popularize the term, but it's also possible that others had used this term before.
In any case, the Basque beret has become a symbol of Basque culture, as well as a popular fashion accessory worldwide.
The Basque beret is typically made of black, red, or blue wool, and is often adorned with a small pom-pom called a "cabillou". Although traditionally associated with the Basque region, the beret has become a popular fashion accessory worldwide, worn by people from all walks of life.