ICE officers can be sworn as immigration officers, customs officers, or both. Traditionally, ICE/ERO officers (who are the deportation squads) are just sworn as immigration officers, while ICE/HSI agents (who are criminal investigators) are sworn as both. I’ve seen some suggestions that ERO might start getting sworn as customs officers as well, although I’m not sure if this has happened.
Immigration officers have the powers Tiger Guy cited, from 8 U.S. Code § 1357. This gives them fairly limited powers without a warrant. They can arrest noncitizens for unlawful presence and can arrest anyone for felonies under the immigration laws, but they can only make these arrests without a warrant if they have reason to believe the person will escape in the time it takes them to get a warrant. They can also arrest for non-immigration crimes without a warrant under the same condition of “the person might escape before they could get one,” but only if they’re performing immigration duties at the time. In other words, an immigration officer doing an immigration raid who sees someone with drugs can arrest them. An immigration officer cannot go out to look for drug deals.
On the other hand, customs officers have broader powers. Under 19 U.S. Code § 1589a, a customs officer can make a warrantless arrest for any federal crime committed in their presence or for any federal felony on probable cause. There’s no “must be performing customs duties at the time” rule, and there’s no “only if there’s no time to get a warrant” rule. The authority of a customs officer is basically the same as most other federal agents.