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My girlfriend was refused a UK visa as she didn't mention that she was arrested in 2010 for shoplifting (but only received a warning). At the time she was told she would have no record and it was a slap on the wrist. Please note that she was NOT convicted of any offence. During her interview, the subject came up and she admitted that it happened but did not think she was in any trouble for it, true, but they still denied the visa.

They cited article 320 (7a) and said that any future applications could result in getting slapped with a 7b 10-year ban if changes aren't made.

If we reapply and say that yes, she was arrested but not charged, is it likely that they will grant the visa, or is the "lie" from the first one going to make it impossible?

Your gf got a refusal under Paragraph 320 (7a)...

(7A) where false representations have been made or false documents or
information have been submitted (whether or not material to the
application, and whether or not to the applicant’s knowledge), or
material facts have not been disclosed, in relation to the application
or in order to obtain documents from the Secretary of State or a third
party required in support of the application.

It's a nasty one but let's have a look at the form to see if they got the refusal right...

Based upon what you wrote, your gf was arrested and got a warning. That counts as a type of penalty. So she should have answered this question with a 'yes' and proceeded to give the details. Paradoxically, since the offence was in 2010 they would consider her rehabilitated by now and her history would have been a non-issue. I assume she answered 'no' to this question. The fact that she got called in for an interview should have set off all her alarm bells; there's paperwork on her somewhere.

Your question...

if we reapply and say that yes, she was arrested but not charged, is
it likely that they will grant the visa, or is the "lie" from the
first one going to make it impossible?

Why the use of quotes for "lie"? What term describes it? If something is asked on the form they want it answered correctly. Let's look at what the Tribunal has to say about it...

When a direct question is asked, and answered untruthfully, there is
both a false representation and a non-disclosure; and it is not open
to an Appellant who gives an untruthful answer to a direct question in
an application form to say that the matter was not material.

Source: The head note from Kenya [2010] UKUT 165 (IAC)

Terminology aside, unwinding these types of situations is terribly difficult and a fresh application where she answers 'yes' will not change her history as someone they don't trust. She will need to have a significant change of circumstances to get them to consider it. If her next application fails, they have the option, but not the obligation, to fix a 10 year ban.

Usually the prescription is a change of circumstances along with the persuasive skills of a solicitor that specialises in unsnarling Paragraph 320 refusals.

So for all practical purposes, the outlook on a fresh application right now is dim.

Your comments...

basically it is not an impossible task to reapply and have it granted,
correct?

Anything is possible. I have seen applications succeed where I was dead certain they would fail. And the reverse! The above still applies, I put a very low chance of success.

What would constitute a "significant change of circumstance".

That's already described here: UK visitor visa refused (multiple sponsors) - second part

Would you suggest speaking to an immigration lawyer just for advise or
to handle the case entirely.

Instruct them to represent the application. If they were regulated in the UK, their answer would pretty much match what's written here. You need representation, and it will be bloody expensive to unsnarl this.

Time is of the essence and we need to get things sorted pretty quickly

Everybody in the world wants to get things sorted quickly. Failing to plan and organize can spell doom for applicants who might otherwise succeed.

Update 20 July 2016

You uploaded your gf's refusal notice. Reading the notice, the situation is actually graver than indicated in your original question. As I suspected above, there is paperwork on your gf that they accessed and she was invited to attend an interview so they could get the whole lurid affair on video.

If they ask someone to attend a personal interview, the person should be seeing red flags and hearing all kinds of alarm bells. Withdraw the application and instruct a UK solicitor right then. And fix your resolve to be utterly transparent.

Her interview performance was especially damning. She is not going to be back for a long time, so I revise my take on it: from dim to 'not going to happen'.

Also I doubt that she will be able to instruct a qualified UK solicitor for a while because the Law Society will question the solicitor's ethics if they gave a client false hopes. In terms of an estimate, if you got the UK's leading practitioner for cases of this type it would run about GBP 4,500 if and when they took your gf under client care (but that's a guess).

Finally, if your gf applies again without representation, the nascent threat of a full 10 year ban becomes impending. Hopefully your question will attract some other answers from people offering a more optimistic viewpoint.

I'm just going to add a very meagre answer compared to Gayot Fow's comprehensive analysis of your situation.

A caution in the UK is not a warning with no consequences that it may at first appear. If people were fully aware of what a caution is, they would probably be recommended not to accept it.

Accepting a caution can seem appealing because it means you will not
go to court for that offence. It’s phrased like it’s a slap on the
wrist, and can seem like an easy way out. However, a caution is an
admission of guilt and can still affect you in the future: it stays on
your record for 5 years and can affect visas, travel, and job
applications.

Now that it is unlikely that your girlfriend can get a visa in the short term, you may wish to consider alternatives. Since the UK is still in the EU, there are family routes to enter the UK. This probably involves marriage though. Unless you adopt her or something perhaps!

As Gayot Fow says, you are strongly recommended to consider legal representation for your options at this stage.

A caution is not a criminal conviction. The thing is a simple caution was not required to be disclosed back in 2010 by the Home Office; even in 2013 I didn't have to disclose it for my extended visa as there was no section related to simple caution in the form back then — until recently in 2016 when you had to declare it on the application form because now there is a section for it.

I also have a simple caution back in 2010 but I was not stupid enough to never declare it for my ILR in 2016.

Your girlfriend will have a tough time as she is now under deception so she may not even qualify for ILR or naturalization.