Al-Alam, an Arabic-language, Iranian state-run television station, showed the woman, 26-year-old Faye Turney, wearing a black headscarf and smoking a cigarette. The footage also showed some sailors and marines eating.
In the video, Turney says, "obviously we trespassed into their waters." She then says she had been well-treated.
"They were very friendly, very hospitable, very thoughtful, nice people," she says. "They explained to us why we'd been arrested. There was no aggression, no hurt, no harm. They were very, very compassionate."
Al-Alam also showed a letter it said she had been allowed to send to her family in Britain. The letter said, in part, "I have written a letter to the Iranian people to apologize for us entering their waters."
The footage also showed a brief scene of what appeared to be British sailors sitting in an Iranian boat in open waters after their capture.
Britain's government reacted by saying that it was "completely unacceptable" for pictures of British sailors to be shown on Iranian television.
(compiled from agency reports)