Extension material that may cut or break the hair and melt with the application of excessive heat is:

Nylon is a synthetic material that can be used for hair extensions, but it has properties that make it susceptible to heat. When exposed to excessive heat, nylon can melt, potentially leading to damage not only to the extension itself but also to the natural hair. This melting can create a situation where the hair strands become weakened, cut, or break, posing a risk to the integrity of the natural hair.

In the context of hair extensions, understanding the characteristics of each material is crucial for making selections that minimize damage. For instance, while Kanekalon is often used for its heat-resistant qualities, yarn and cotton are not typically associated with the risk of melting. Therefore, nylon is distinctly known for its vulnerability to heat, making it the correct answer in this scenario.