One more updated pattern. Not really carbon fiber, but it’s the most popular pattern, so I’ll give you an extra choice.
Source Atle Mo
From a drawing in 'At home', J. Sowerby, J. Crane and T. Frederick, 1881.
Source Firkin
A seamless pattern from a tile made from a jpg on Pixabay. To get the tile select the rectangle in Inkscape and use shift-alt-i.
Source Firkin
A bit strange this one, but nice at the same time.
Source Diogo Silva
This is a remix of "blue wave-seigaiha".I hope this subtle color version of Seigaiha would be suitable for background .
Source Yamachem
Remixed from a JPG that was uploaded to Pixabay by susanlu4esm
Source Firkin
Light and tiny, just the way you like it.
Source Rohit Arun Rao
Prismatic Hypnotic Pattern No Background
Source GDJ
And some more testing, this time with Seamless Studio. It’s Robots FFS!
Source Seamless Studio
The image depicts a seamless pattern which includes hexagonally-aligned gourds with BG in light-brown.
Source Yamachem
From a drawing in 'Artists and Arabs', Henry Blackburn, 1868
Source Firkin
Not the most creative name, but it’s a good all-purpose light background.
Source Dmitry
Prismatic Groovy Concentric Background 2
Source GDJ
Uses spirals from Pixabay. To get the basic tile select the rectangle in Inkscape and use shift+alt+i
Source Firkin
Small gradient crosses inside 45-degree boxes, or bigger crosses if you will.
Source Wassim
Design drawn in Paint.net, vectorised using Vector Magic and finished in Inkscape.
Source Firkin
From a drawing in 'Az Osztrák-Magyar Monarchia irásban és képben', 1885.
Source Firkin
Zero CC tileable wood texture, made by me procedurally in Neo Texture Edit.
Source Sojan Janso
Tile-able Dark Brown Wood Background. Feel free to use it as a background image in your designs or somewhere on the web. By the way, the color seems to be close to Coffee Brown.
Source V. Hartikainen
Not a flat you live inside, like in the UK – but a flat piece of cardboard.
Source Appleshadow