Many sources claim that Bill Wyman's first fretless bass was 'Japanese' or 'home made', amongst other misinformation or forgetfullness. More recent research says that it was originally a 1959 Dallas Tuxedo bass, made in the UK by the Dallas Arbiter company.
Bill bought it second hand, and took this light-weight, solid-body, single pick-up, short-scale instrument and heavily modified it by removing the frets (some sources say the frets had already been removed), re-shaping the body (adding an upper cutaway, shortening the body behind the bridge, narrowing it under the two control knobs, and moving the output socket), painting it red, and making a black scratch-plate.
He played this bass in The Cliftons before he joined the Rolling Stones. Later he added a second pickup near the bridge, added a third control knob, painted it black, and made a red scratch-plate. In 2011 the UK company 'The Bass Centre' issued their own fretted version for sale, called the 'Wyman Bass', based on the two-pick up fretless version. Here are some pictures!
Dallas Tuxedo bass
Bill's customised Dallas Tuxedo bass in it's single pick-up configuration, the body painted red.
The same bass in the later two pick-up configuration, with the body painted black.
Early photo of Bill playing the customised bass, red body.
2011 'Wyman Bass' by The Bass Centre, UK.