Cox's Bazar has had a substantial ethnic Burmese and Arakanese community for generations, and the Burmese Market near the town centre reflects this. The stalls sell handwoven textiles, longyi (Burmese wrap sarongs), lacquerware, sandalwood products, dried goods, and household items more likely to originate in Myanmar than Bangladesh.
Prices are negotiable. The market is compact and the goods are not unique — you can find similar items in Yangon at lower prices — but for travellers who won't be crossing into Myanmar, it offers a genuine window into a different material culture. The vendors are generally relaxed about visitors looking without buying. Go in the morning before the heat and the tourist groups.