Over the last few weeks, our garden is being overrun by this one type of
weed. Its a ground cover type for sure, and jeesh it has so many roots and
tendrils underground that my first thought was bindweed. But after searching
the various weed identifying sites maybe not.

Ground elder. Hit it with glyphosate spray on a nice still day wait
about 3 weeks and then dig as much out as you can (complete clearence
if possible on a small area). Repeat ad infinitum. Leave just a 1cm
piece in the ground and you will have a new plant. However, if you
weaken it with glyphosate first it is less inclined to regrow. It is
quite tricky to eliminate from a border with perrenial shrubs. (easy in
open ground)

I hit my ground elder with weedkiller whenever I have a batch going.
There is no chance of eliminating it as there is a field full just
beyond my boundary fence. All I can do is halt it at the borders.

The other way is to blitz it out of existance by removing all the
leaves with a mower or strimmer at least twice a week. Allow it to see
daylight and the roots store massive reserves of energy for later.

I'd opt for a multi purpose weed killer but the garden has well established
conifers down both sides which these weeds are swamping and I'd sooner not
kill those through ignorance.

A long wand sprayer and a steady hand on a still day and you should not
cause too much collateral damage. If you are worried put a plastic
sheet up against the plant you want to protect. And if you make a
mistake prune off the part of the plant hit with glyphosate. It has to
hit green leaf to do any damage. The exceptions being very waxy plants
like ivy and holly seedlings that will usually survive.

Remember that grass is exquisitely sensitive to glyphosate so don't get
spray on your shoes and walk across a lawn!

Regards,
Martin Brown