Is there really any proof of this, or is it just annoying fan speculation?
One non-canon creation by the fans, is something called a "warp sustainer." No existence on the show.
Ships and object in Star trek are depicted as being able to travel at warp speed without a warp engine. Just rewatched Brothers, Picard said the saucer would continue for two minutes at warp speed after separation, the saucer has no warp drive.
The probe that delivered K'Ehleyr to the Enterprise would appeared to have been too small to possess a warp drive. It was a seemingly empty shell containing the ambassador. It traveled at fairly high warp speed.
In TOS, the phaser beams traveled (for some distance) at warp speeds.