A glyphosphate based weedkiller, such as Roundup in the UK, is probably the best. However, you will have to be very, very careful not to get it on any of your other plants or it will kill them also.
A safer option would be to clear all the plants out of one area of your garden at a time and give it a good digging over to remove all of the vinca roots.

As suggested by Bulbaholic, a glyphosphate-based weedkiller will sort the problem out. At one time you could buy a red gell form but I think it has been withdrawn (another EEC directive I suppose). As your climate is rather warmer than ours, try painting it with a mixture of glyphosphate and wallpaper paste now and then again in the springtime. It is just a matter of patience but don't try pulling it out, otherwise the roots will just break off.

Thank you Mr B. & Jimmy. I feared the "dig it out" advice - I have been doing this for the past 2 years and I just cannot get all of it, so back it comes, which is why I, reluctantly, am looking to go down the weedkiller route. It's unfortunate that the bed is shallow soil over fragmented rock, and try as I might, roots find their way into places I can't reach. We CAN, apparently, get Roundup here, but it's difficult to get hold of, so I may have to register as a professional agriculturalist/horticulturalist first! I don't fancy trying to do that! Maybe a bottle of concentrate smuggled back in the suitcase next time we're back in the UK.
Do you know of any other proprietary brands which contain glyphosphates? I believe the Monsanto patent has now expired. I'd never thought of the wallpaper paste idea, Jimmy - sounds worth a try.

Just buy glyphosate concentrate rather than round up , works just as well and is much less expensive. It kills through the foliage, so if you inadvertently got the spray on the woody bases of other plants, it won't kill them.

Oh, Bamboo, thank you: that is more the sort of advice I had hoped for. Is it readily available through ordinary garden centres, do you know? I really don't like using chemicals in the garden, and hence know very little about what's available and what each one does, but I am at my wits' end. I thought maybe using Roundup near the woody stems of our established shrubs and trees might see them off. We have maybe 25 square metres of steeply sloping rockery covered with it, and I have already torn my hands to shreds trying to prise it from round the base of a huge rose bush. It doesn't matter how deeply I dig, there are always more roots.

Not sure about garden centres, but I don't see why not - I bought my last glyphosate in Wilkinsons, their own brand. Best used when the plants are growing strongly, and the ground is damp, and no rain is forecast for 8 hours - spray all leaves and stems. You'll need to repeat more than once...

I shall send OH back to the UK with a big suitcase! Thank you!

And a large pump sprayer, if you haven't got one already...

Got one of those, thx!

I thought you probably had, lol!

When I have something like this and want to kill it, I paint it a glyphosphate based weedkiller and then put the long stems in plastic bags and seal it with a tie. That stops it being washed off and waving about and touching anything else.

Is it wise to let OH loose with glyphosphate Gattina... Remember the dodgy Cherry Tomato the other day ?
I was going to suggest a flame gun, then remembered your blog about lighting bonfires.

Oh dear, Denise, I'd have to spend WEEKS painting each individual stem, and there wouldn't be enough plastic bags in the whole of Bologna! I thought it might be easier to isolate what I DON'T want to kill with plastic, and spray the rest.
Good Gracious, Pimpernel, do you think I'm mad? I wouldn't let OH loose with so much as a bag of smarties.......... He climbs ladders and lights bonfires (what is it with blokes and bonfires?), I do the difficult and dangerous stuff and anything that involves measuring and mixing and spraying and planting.
I have a useful little phrase which covers a multitude of situations:-
"Gli dico cos'a fare, e lui lo fa!" (I tell him what to do, and he does it) It has all my female Italian friends laughing their heads off in agreement.

Sounds like a recipe for a good marriage to me,Gattina...

LOL! Works for us. Well, it works for me. I've told OH it works for him, too, so he believes me.

;-)

It actually works well in most households, Gattina. Let the woman take the blame when it all goes horribly wrong!!!!!!

Ah, Mr B., but when women are properly in charge, and the men listen and do exactly what they're told, it NEVER goes wrong! ;-)))))

:-)))