Panama hats originated in Ecuador. 

(When I first heard this I did wonder if  it was one of those regional fights like Cornish Pasties, but a quick Wiki told me that straw hats woven in Ecuador, like many other 19th and early 20th century South American goods, were shipped first to the Isthmus of Panama before sailing for their destinations in Asia, the rest of the Americas and Europe. For some products, the name of their point of international sale rather than their place of domestic origin stuck, hence “Panama hats.”)