General Discussion

https://www.aclu.org/know-your-rights/border-zone

You have the right to remain silent or tell the agent that you’ll only answer questions in the presence of an attorney, no matter your citizenship or immigration status. You do not have to answer questions about your immigration status. You may simply say that you do not wish to answer those questions. If you choose to remain silent, the agent will likely ask you questions for longer, but your silence alone is not enough to support probable cause or reasonable suspicion to arrest, detain, or search you or your belongings.

A limited exception does exist: for people who do have permission to be in the U.S. for a specific reason and for, usually, a limited amount of time (a “nonimmigrant” on a visa, for example), the law does require you to provide information about your immigration status if asked. While you can still choose to remain silent or decline a request to produce your documents, people in this category should be aware that they could face arrest consequences. If you want to know whether you fall into this category, you should consult an attorney.
Generally, an immigration officer cannot detain you without “reasonable suspicion.” Reasonable suspicion is less robust than probable cause, but it is certainly not just a hunch or gut feeling. An agent must have specific facts about you that make it reasonable to believe you are committing or committed, a violation of immigration law or federal law.

If an agent detains you, you can ask for their basis for reasonable suspicion, and they should tell you.
An immigration officer also cannot search you or your belongings without either “probable cause” or your consent. If an agent asks you if they can search your belongings, you have the right to say no.

An immigration officer cannot arrest you without “probable cause.”
That means the agent must have facts about you that make it probable that you are committing, or committed, a violation of immigration law or federal law.