The way I read it, if you have been suspected of a public offense, then the officer may demand your name and address, but not necessarilly an ID card. Here are the laws for your review.
And if you fail to provide that info...77-7-15. Authority of peace officer to stop and question suspect -- Grounds.
A peace officer may stop any person in a public place when he has a reasonable suspicion to believe he has committed or is in the act of committing or is attempting to commit a public offense and may demand his name, address and an explanation of his actions.
Don't forget this gem...76-8-301.5. Failure to disclose identity.
(1) A person is guilty of failure to disclose identity if during the period of time that the person is lawfully subjected to a stop as described in Section 77-7-15:
(a) a peace officer demands that the person disclose the person's name;
(b) the demand described in Subsection (1)(a) is reasonably related to the circumstances justifying the stop;
(c) the disclosure of the person's name by the person does not present a reasonable danger of self-incrimination in the commission of a crime; and
(d) the person fails to disclose the person's name.
(2) Failure to disclose identity is a class B misdemeanor.
And this one...77-7-16. Authority of peace officer to frisk suspect for dangerous weapon -- Grounds.
A peace officer who has stopped a person temporarily for questioning may frisk the person for a dangerous weapon if he reasonably believes he or any other person is in danger.
77-7-17. Authority of peace officer to take possession of weapons.
A peace officer who finds a dangerous weapon pursuant to a frisk may take and keep it until the completion of the questioning, at which time he shall either return it if lawfully possessed, or arrest such person.