Iran said Tuesday it had formally protested the F.B.I.’s arrest of an American newscaster who works for the Iranian government’s Press TV, and her family said rallies in Washington and elsewhere were planned if she was not freed.The arrested American, Marzieh Hashemi, has been held for more than a week as a material witness in an unspecified criminal case and has appeared before a grand jury in Washington twice. She has not been charged with a crime.Iranian officials have seized on the arrest as a provocative new irritant in the tense relationship between Iran and the United States, which has worsened since the Trump administration renounced the Iranian nuclear agreement last May and reimposed severe sanctions on the country.Ms. Hashemi, 59, who has lived in Iran for more than a decade and travels back to the United States sometimes to visit family members, was arrested Jan. 13 in St. Louis and transferred by the F.B.I. to Washington.Under American law, witnesses can be arrested if the government can show that their testimony is “material to a criminal proceeding” and that they might flee. The law generally requires the release of such witnesses after their testimony is completed.One of Ms. Hashemi’s children, Hossein Hashemi-Niasari, a graduate student at the University of Colorado, said by telephone that she “hasn’t been treated very well” in American custody and was scheduled to testify a third time on Wednesday.If she is not released, he said, “there are going to be fairly large protests by the end of the week” by Black Lives Matter and other groups.Ms. Hashemi was born into an African-American Christian family in Louisiana. Originally named Melanie Franklin, she converted to Islam after the 1979 Islamic Revolution in Iran. According to Iranian news accounts, she moved to Iran in 2008 and became one of its most recognized English-language news broadcasters.Iran’s Foreign Ministry summoned the Swiss ambassador in Tehran to formally protest the arrest and demand Ms. Hashemi’s release, Iranian news agencies reported on Tuesday. Switzerland represents American interests in Iran because diplomatic relations between the United States and Iran have been severed since 1980.The protest came a day after Foreign Minister Mohammad Javad Zarif of Iran, using the Martin Luther King Jr. holiday in the United States as a backdrop, accused the American authorities of racism in their treatment of Ms. Hashemi.“The US govt needs to explain how Marzieh Hashemi — a journalist and grandmother — is such a flight risk that she must be incarcerated until she finishes her testimony to a grand jury,” Mr. Zarif said in a Twitter post. “50 years after MLK assassination, US still violates the civil rights of black men and women.”The F.B.I. has declined to comment on the case. Ms. Hashemi’s court-appointed lawyer in Washington, Preston Burton, said he could not comment.Ms. Hashemi’s arrest came less than a week after Iran acknowledged that it had arrested a United States Navy veteran last July on unspecified charges. The veteran, Michael R. White, of Imperial Beach, Calif., was the first American known to be imprisoned by Iran since President Trump took office two years ago.At least three other Americans are known to be in Iranian custody, including two of Iranian descent, on sedition or spying accusations that they have denied.Mr. Zarif and other officials in Tehran have often complained about what they call the unjustified imprisonment of a number of Iranians by the United States.