Bill staggered Cyril, showing fibre (7)
I believe the answer is:
acrylic
'fibre' is the definition.
(acrylic is a kind of fibre)
'bill staggered cyril' is the wordplay.
'bill' becomes 'ac'.
'staggered' indicates an anagram.
'cyril' is an anagram of 'rylic'.
'ac'+'rylic'='ACRYLIC'
'showing' is the link.
(Other definitions for acrylic that I've seen before include "The synthetic fibre is icy, Carl" , "Textile fibre - icy Carl (anag)" , "Type of twenty Dn" , "Pungent acid" , "fibre that's synthetic" .)