No, it won't affect the soil.
To avoid using chemicals, remove top growth every time you see it.
Continue until you don't see any more come up

Thanks Jean, that sounds simple. I'll do just that!

And consider that if you use chemicals, there is a chance that some of these runners may still be linked to the main plants.

Thanks drmbear.

I want to remove - eradicate - the raspberries completely (they didn't do well where they were anyway), not just the runners, so would be happy for the main plants to leave too were I using chemicals, but removed the main plants some time ago, leaving just the occasional runner.
The chemical option doesn't appeal to me, particularly.

Coming to an organic gardening forum ans asking about using any glyphosate product is not a good thing to do since any glyphosate product is not acceptable to an organic grower.
For an organic grower to remove unwanted plants if digging them out is not an option then keeping them cut down would be the next step and covering them with a good heavy mulch would also work. Vinegar might also be an option.
But no synthetic plant poisons.

Whenever you see a raspberry sprout, tug it a bit to get as much as you can out of the ground without wrecking the tomatoes, then clip it off at ground level.

Eventually they will give up.

============
If you need to kill something mixed in with a desirable plant, you can selectively apply herbicides.

Google for "The Tongs of Death" or "The Tongs of Doom". It's a home-made sponge applicator for these situations.

To keep kimmsr from having a conniption, let's pretend you are using 100% pure agricultural vinegar, OK?

Thanks lazygardener, very much, for the tips. Kimmsr, I didn't know what I was saying - I'm as organic a gardener as a person can possibly be, honest. Desperation can do funny things to a man. Fear not, all is well. I've come to my senses. I'm going the pulling out / 100% hospital strength vinegar route!

Why are there still people on this forum suggesting chemicals. Why aren't they banned? I stopped reading this ages ago because of this.
There are so many great organic methods for all aspects of horticulture.
And there are plenty of forums where roundup is fine to bring up.

I think moderators need to WARN, then BAN. Clean this up!

Fireweed,
It seems to me that perhaps it would be better to try and convert the non organic into organic.

I'm not really in the tolerant state of mind, and when I read advices ending with "do as you like, it's your garden", I think : your garden, in our future on our planet with finite borders.

So it seems to me that when a chemist gardener comes on an organic forum, it would be better to try and explain why organic is the way, for him, for his garden, for his kids, and for ours.

Geeze, folks........chill!!

First, other than personal attacks, religion and politics, there are NO topics that are off-limit enough on any forum that they or their presenter should be banned.

Second, no one was advocating the use of a chemical herbicide (Round Up). If you read carefully, the OP clearly stated that was not a preferred choice and asked what the forum could suggest as an organic alternative. And any further references to RU were to answer a specific question or provide support why this method was not necessarily a wise choice.

As to the proposal of discussions concerning Round Up being banned from this particular forum, I think that is counterproductive. Many visit this site because they are moving away from chemically enhanced gardening methods and like the OP, have legitimate questions about relative effectivess of alternative organic methods. Any discussion that increases one's knowledge and awareness is a valid and important contribution regardless of what products are brought up.

To be perfectly honest, hard-nosed, gung-ho organophiles who loudly condemn any mention of non-organic methods can be extremely off putting to new forum participants. Let's exercise a little tolerance, okay?

I stopped reading this ages ago because of this.

Oh really?

Is that a hint LG?
I love all aspects of gardening and read a number of forums here. But yes LG
I don't understand why it's ok for anyone to give chemical options on an ORGANIC forum, when OP clearly stated they didn't want to use poison.