1 Theophilus: Tally ho! I'm off to make some more hats.
2 Theophilus: I've got a couple of Panamas I need to weave out of straw before retiring for the night.
3 Ophelia: Do you often work so late?
4 Theophilus: Oh yes! No rest for the wickerwork!

Wikipedia says wicker is a word used to describe furniture or baskets made using a weaving technique. Specifically, it says, "As such, it is not an actual material, but a technique." And if you use Google to search for "wicker hats" you get lots of stuff about straw hats, which are woven in a similar manner to wicker furniture and baskets, just out of lighter material.

Although not always. The Danes in particular are known for their use of heavier materials to make woven headgear. Because as we all know: Something is rattan in the straw-hat of Denmark.