Probably not.
If it's a modern Hybrid Tea type rose (& sounds like it might be) it's probably grafted onto a rootstock rose--that is, a rose that forms better roots & which is used to feed the top part of the plant--grafted onto the rootstock--from a rose that has prettier flowers.

Rootstocks aren't famous for their beautiful flowers--that's not their purpose--& sometimes they send up little shoots like you describe. They're called "suckers"

I can't say for sure, it's just a guess. Some roses aren't grafted & so are called "own-root". Any suckers they send up are going to be the same as the flowering part.

Why don't you ask your neighbor if they will try to root some of the rose for you, or let you try? Or if it's a modern rose that's still patented, they can tell you the variety & where to get one. Most rose lovers would be complemented & love to share.

Modern roses can be slow to root, but try several ways, this fall, or again in late spring, if you don't have success. Lots of threads here with detailed instructions.

She doesn't know anything about the roses, she said they came with the house. She knows to trim at the first set of 5 leaves and prunes them way back in January. No food or anything. They're about 5 feet tall and 4 feet across. Each rose comes on a long stem. All I know about roses is what's a floribunda and what's not. My only rose is a white Iceberg. So there's no point in digging up that small plant? I thought a sucker means a branch attached to the main plant? This came up in the dirt away from the main bush. It's dirt, not soil, so roses must be survivors !

Why don't you post a photo in the "Roses" forum? Folks there are very good at identifying from pictures (I just posted one).

If you want to try digging up the sucker & it's okay with the neighbor--go for it!

But from your description of the rose, I'm guessing (& it's just a guess) that it's a hybrid tea & most all of them are grafted. A real sucker IS from the main plant, but it's probably coming from the ROOTSTOCK of the plant, not the grafted top part with the pretty flowers.

But it could also be a seedling--not all roses are fertile & not all seedlings are pretty or look like their parents. Or your neighbor's rose could be an own-root plant (not grafted onto a rootstock), so a sucker from it would look like the blooming parent--it's impossible for me to know--these are the 3 possibilities: a seedling, an own-root sucker, or a rootstock sucker.

If you dig it up, get as much root as possible & keep it well watered & in the shade until it recovers.

(Iceberg is a fine plant--one of the best white roses of any kind ever. I love it)

Thank you both. I guess I won't dig it up as it sounds like it's from the rootstock. Rats.

Visit with your neighbor a bit & find out when she moved into her home. If she moved there in, say, 1980 & the rose was already there, well, you know the plant is absolutely no more recent than 1980.

Also might look closely around the plant's base to see if there is still an old name tag attached or buried underneath nearby.

Take as clear a photo as you can of a fresh bloom, before the sun changes it much--lots of folks post photos taken from a phone. Another picture of the general plant itself, its canes (does it have lots of thorns, or just a few?) is useful. Also, smell it--is it fragrant? Some lovely roses are not.
Does it bloom repeatedly through the year?--some only bloom in spring, but HTs repeat.

Get the pictures on the Roses forum with a question: "what rose is this?" & the information you know. I bet you'll get a correct identification & where to buy it. It's like a puzzle--little pieces that make up the whole picture.

For example, just your description made me first think: Katherine T. Marshall, a popular old hybrid tea with single blooms to a stem, a pure light pink, a light fragrance. (Just a very wild suggestion of what it could be, nowhere near enough information to say). Next thought was Belinda's Dream--a more recent pink rose, very good bloomer, very widely planted in my area--fragrant, medium pink, quite a few petals.

You can look at photos & read descriptions of roses suggested on a giant database called "Help Me Find". It even lists nurseries where to find each rose.

Here is a link that might be useful: Help Me Find roses search page

OK, I'll root around for a tag and ask her when she moved in. If I take a picture with my phone, how do I post it right on the thread and not have to do the photo bucket thing?

Uhmmm, I'm pretty new at it, but:

The file size of photos is limited, something like 5000KB ( I dunno, so if it won't post, do this:

file the picture on your computer & select an editing program, like Paint, or whatever, to reduce the file size, like make the photo 400x 400 dpsi or whatever.

Type in your response like you've been doing.
Under "Post A Follow-Up", there's "Image file to upload" & a "Browse" button.

Click "Browse" & upload the photo file, click "preview" & then "post". If the image file is not too big it will go through. If not, it will tell you (& also the size limit, I don't recall what it is exactly).

The proper instructions are somewhere on GW, you can search "how to post a photo" if this doesn't work.

Thanks ! I'll try it.

I posted another thread, asking for a how-to on rooting a cutting. Thanks all.