When looking at the hanja for '남' on my Windows IME I was surprised to see '바구니 남' as the third entry. I couldn't find a Korean word that uses 藍 / '남' to mean 바구니 / basket. Are there any examples of Korean words that use this character?
It appears to be a mistake. There are two similar characters:
藍 - 쪽 람 - means indigo - found in the word 남색 (indigo colour)
籃 - 바구니 람 - means basket
The difference between these can be difficult to see, but the radical is different - 쪽 람 has the "plant" radical 艹, and 바구니 람 has the "bamboo" radical 竹.
Both of these will be pronounced 남 when found at the beginning of a word.
Using the Windows Korean IME, typing 람 + 한자 key gives 藍 쪽 람 as the 3rd choice, but typing 남 + 한자 key gives 藍 바구니 남 as the 3rd choice - it appears the two characters got mixed up here.