Do not lie to a federal officer. An ICE officer is a federal officer. Lying to a federal officer could constitute criminal obstruction of justice which could be considered an aggravated felony. A criminal conviction (such as an aggravated felony) is a potential ground for removal from the United States. The ICE officer has access to databases and other information to potentially determine your identity and your status, or lack of status, in the United States. A false claim of citizenship is one of the most serious offenses in the eyes of immigration, and will lead to deportation with no waiver available at all. You are better off saying nothing than telling a lie.
Keep important documentation with you (preferably not your passport). If you have a work permit or any other document from immigration authorities, always keep it with you in case you are stopped by ICE. If asked for your work permit, present it to the officer. If you have any documentation for a local address, keep it with you. If you obtained a green card, make sure you have it handy or else you could be subject to a $500 penalty along with the hassle of ICE trying to confirm your identity. Proof of residency, such as a state ID, may be critical if a bond determination is made by an Immigration Court.