No, everyone does. Even ICE wouldn't argue that somehow the 5th and 6th and 14th Amendment only applies to citizens because that reading doesn't give them unlimited power to deny due process, but rather, it would place non-citizens out of their jurisdiction entirely. Although that would also conflict with the plain wording of the constitution itself and also, the drafters of the 14th Amendment have testified that their intent was to cover all people. Their argument had always been that as an administrative agency even though what they do is virtually indistinguishable from the exercise of judicial powers under Article III, they do not serve Article III courts and therefore, detention and removal are not "punishment", but an administrative matter. Otherwise, we'd have a public defender system at the very least for immigration cases, because Gideon v. Wainwright would apply under cases and controversies arising out of Article III powers and the exercise thereof. Instead, by explicitly not basing their actions on Article III powers, they've contended that the protections in the constitution doesn't kick in. This is not a winning argument but enough of their rationale have managed to stand. The 4th Amendment actually still applies except without a lawyer one can hardly seek redress, especially when ICE can remove you beyond the court's jurisdiction at will. ICE obviously wants its cake and eat it too, but since that doesn't tend to fly in court, even the Court during the first Trump admin, the only way they can thread the needle in any way that is even remotely persuasive would necessitate that they'd have jurisdiction over all who are present, since Article II powers delegated by Article I also requires the Constitution to establish personal jurisdiction over everyone in the country.