One question: Is this board pine? Many cheap flooring materials are not intended for direct traffic, people remove the carpets and polish boards that can't stand up to the direct pressure of the wheels of a chair for example. Also there may be rot or termite or borer damage weakening the boards, as a landlord I would get it repaired as soon as possible to establish the cause and maintain good will, which is worth more than a floor board.
As a legal point it sounds like your tenant is exacerbating the problem out of the mistaken belief you need to provide hard floors for their use however, so I would say any damage caused to sound boards subsequent to written advice to take steps to prevent it is the tenants responsibility should this issue go to tribunal, open and shut.