[texas booster]

Pretty sure yes, and that’s the idea behind this test flight.

In all fairness, Apollo 13 was probably the first to circle the moon and come back to Earth.

[ProtectOurFreedom]

That's why I specified "unmanned." However, before Apollo 13, there was Apollo 8:

NASA --> The success of earlier Apollo flights, problems in the development of the lunar module and concerns that the Soviet Union might be ready to launch astronauts around the Moon led NASA to change the flight plan for the Apollo 8 mission from an unpiloted, Earth-orbiting mission to a crewed flight around the Moon.

Frank Borman, Jim Lovell and Bill Anders were the first crew to fly atop the powerful Saturn V booster, ultimately spending 20 hours orbiting the Moon. On Christmas Eve, 1968, the crew gave a memorable reading from the Book of Genesis, and while in orbit Anders took the iconic "Earthrise" photo.

Crew:
Frank Borman, commander
William A. Anders, lunar module pilot
James A. Lovell Jr., command module pilot

Launch: Dec. 21, 1968
Splashdown: Dec. 27, 1968

Many of us remember that reading from Genesis from the moon like it was yesterday. That was absolutely the most thrilling thing ever, especially in the midst of the space race with Russia, the threat of nuclear war, and only 23 years after WW II ended. It also helped bring the nation together after the horrible race riots of the few previous years.

[NorthMountain]

Apollo 8 circled the Moon and returned to Earth, but with a crew.