The most common and most likely answer to this crossword clue is the 7 letter word TAFFETA
I've seen this crossword clue in Independent
T
A
F
F
E
T
A
More clues leading to the result TAFFETA
- Lifted plump cheese to find crisp material
- Material from Falstaff et al.
- Fabric is fine to iron later around the edges
- Grease spilt over cheese cloth
- Obtaining material, staff head missed tea break
- Material round round cheese
- Cloth put round oily cheese
- Thick-backed cheese material
- Material upset big cheese
- Offensive article about two females going after adult material
Buddy explains!

The straight part
We know that "Material" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case material = taffeta.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does TAFFETA mean?
TAFFETA noun- A crisp, smooth woven fabric made from silk or synthetic fibers.