A narrow stripe that turns up in chenille (4)
I believe the answer is:
line
'a narrow stripe' is the definition.
(a line is a strip or stripe)
'turns up in chenille' is the wordplay.
'turns up' is a reversal indicator.
'in' indicates the answer is hidden within the clue.
'enil' can be found hidden inside 'chenille'.
'enil' backwards is 'LINE'.
'that' is the link.
(Other definitions for line that I've seen before include "classified information" , "business" , "cover on the inside" , "Winter sports slope" , "Long narrow mark" .)