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"...science fiction is sort of like a sociological genome. It's a huge range of possible futures, most of them useless; some vital. You never really know in advance."
- Peter Watts

Faster-Than-Light  
  Describes something that exceeds the usual speed limit on physical objects of 186,282 miles per second in vacuum.  

This is the first use of the phrase in science fiction; as far as I know, it is the first use of this phrase to describe the movement of a real object.

As Earth's faster-than-light spaceship hung in the void between galaxies, Arcot, Wade, Morey and Fuller could see below them, like a vast shining horizon, the mass of stars that formed their own island universe. Morey worked a moment with his slide rule, then said, "We made good time! Twenty-nine light years in ten seconds! Yet you had it on at only half power...."

Arcot pushed the control lever all the way to full power. The ship filled with the strain of flowing energy, and sparks snapped in the air of the control room as they raced at an inconceivable speed through the darkness of intergalactic space.

From Islands of Space, by John W. Campbell.
Published by Amazing Stories in 1931
Additional resources -

In his 1928 novel Skylark of Space, 'Doc Smith describes a ship that exceeds the speed of light, but does not use the phrase "faster-than-light":

For forty-eight hours the uncontrolled atomic motor dragged the masterless vessel with its four unconscious passengers through the illimitable reaches of empty space, with an awful and constantly increasing velocity. When only a few traces of copper remained in the power-plant, the acceleration began to decrease and the powerful springs began to restore the floor and the seats to their normal positions. The last particle of copper having been transformed into energy, the speed of the vessel became constant. Apparently motionless to those inside it, it was in reality traversing space with a velocity thousands of times greater than that of light.

Compare to FTL from The Enchanted Forest (1950) by Fritz Leiber, which is the first use of the acronym.

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Additional resources:
  More Ideas and Technology from Islands of Space
  More Ideas and Technology by John W. Campbell
  Tech news articles related to Islands of Space
  Tech news articles related to works by John W. Campbell

Faster-Than-Light-related news articles:
  - NASA's Warp Speed Starship Design

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