If you don't provide anything for the clematis to climb up and over it will grow across the ground, how good it will look is a different matter.

There are some herbaceous Clematis used for ground cover. Taylors have a few but there are other people who grow them. You might even find some at the local garden centre.

Trouble with even the herbaceous ones (quite a few of those too!) and the climbers grown horizontally (Treasures of Tenbury Wells have lots grown like this) is that they do not grow thickly enough to really cover the ground enough to exclude light from the soil underneath. That cover is what you need from a ground covering plant.
Also most of the ones suitable for this style are deciduous, so no cover in Winter.

Thank you all for your answers. So some types of Clematis could be grown flat for seasonal decorative purposes but as far as weed suppression it would be a fail.

Yup no use at weed suppression at all...

Winter Beauty is evergreen and tolerant of going up, across, down or wherever, is generously vigorous, and has lovely scented white waxy bells in winter. The flowers are weird in that they form in every leaf axil of the string, and it is intriguing following the strand back to it's source.They generally say to allow to scramble through low growing shrubs, like f.ex low growing eunonymus variety, which would probably work.

I have a narrow border of herbaceous paeonies with clematis x triternata rubromarginata scrambling over them.
This provides a second season of interest after the paeony flowers have finished.