The Clarets have had a successful season in Division One, following their promotion last year, and still have an outside chance of reaching the play-offs.
Centre back Cox has played his part and he has decided to concentrate on domestic football, rather than pursuing his international career with Trinidad and Tobago.
"I am delighted that Ian has selected club over country," manager Stan Ternent told the official Burnley website. "He is an integral part of my squad and an excellent footballer."
Meanwhile, Ternent confirmed that four reserve team players are looking for new clubs, as they will not be offered new contracts.
John Boardman, Michael Devenney, Chris Scott and Alex Kevan are all on their way out of the club.
"I am in the process of finding football clubs for the lads," he continued. "They have served Burnley with distinction.
"They all have a future in the game and I would like to wish them every success for the future."
