The club have also been given a suspended £1,000 fine with Clarke's ban, along with the fine, being held until the end of the season.
Clarke and Kilmarnock were both found to be in breach of Rule 77: "Failing to act in the best interests of Scottish football."
The fines come after Clarke said on August 31 he was "disappointed but not surprised" by the decision of the Scottish FA's confidential panel after Dicker was found guilty of series foul play over a challenge on Callumn Morrison.
He also claimed the decision was "pre-judged" after the SFA refused to overturn Dicker's red card shown by referee Willie Collum.
Clarke was in attendance at the Scottish FA disciplinary hearing at Hampden Park on October 25.