The roots can go down and sideways under things several feet. In the bottom pic, what is shown is just what I was able to dig and pull up.

It is not hedge bindweed, with which I am sadly familiar, but it may be field bindweed, which is just as big a thug. If that is the only spot that you have it in, then it is hopefully a newish plant without the large rootstock they can develop. Ready to spray Round Up will do nothing, as you have already learned, so get some RU concentrate (Home Depot, someplace like that). Using your bottom picture as a model, I would try painting the straight round up on the leaves of the "source" plants like those two, and cut off the vining part that has no roots. Be careful and use gloves, please, when using the RU. You could use any kind of brush; maybe one of those foam rubber ones. You will probably have to do it again, as bindweed is awful, but if, as I said, that's the only place you have it, you should be able to get rid of it before it affects the plants it is near.

I wish you all the luck.

I wasn't familiar with hedge bindweed, specifically, so I Googled it. I agree it isn't the hedge bindweed you mentioned but the flowers are very similar.

That's the biggest clump of it in the bed but unfortunately it is coming up in a variety of

places in the bed, and it is all over in the lawn next to the bed. I've tried pulling up some of it by hand but quickly found that doesn't work very well in a cactus bed because of the spines.

After some searching I read some people have seen good long term results from lawn weed and feed products. I wouldn't put that in my cactus bed, of course, but just putting it on the lawn next to the flower bed might make a difference considering how the roots like to travel.

I'll look in to getting some RoundUp concentrate.

I have not tried this, but others in the past have mentioned placing the concentrated roundup in a bowl or cup, and placing a cut end of the plant inside the bowl for the plant to transport the chemical thoroughly throughout the plant. I read this is a very old thread about field bindweed, and the thread's author reported that it was successful.

Looks like Field Bindweed, Convolvulus arvensis. It is smaller in all its parts than Calystegia sepium, Hedge Bindweed. The stem and foliage have a different texture and the flowers don't have the prominent two bract calyx of HB.

I smeared some concentrated glyphosate on the leaves. That's certainly having an effect. I thought of trying to soak the leaves in some small containers of glyphosate + water mix or concentrated glyphosate but I'm not sure I want to leave small containers of nonselective plant killer lying around with a cat here and children occasionally visiting. No telling what they might do with it. Plus, right now most of the weed sprouts in the bed aren't long enough to stick in a cup.

Kind of reminds me of moneywort, though I'm sure it's not. Too bad the flower isn't open.

I tried posting the following update last night but the Houzz servers were having issues.

The vines in the pic in my opening post are dead. I'm not sure how far down they've died but I'm impressed by the results. There are a few other stray vines in the bed that weren't in the pic. Some of those vines are dying and some are still OK. I'll try applying more concentrated glyphosate in about a week after the rain stops.