Limited Stay but difficult to travel

On a sensitive matter such as this, you should talk to a qualified immigration professional and not rely on the advice of random strangers on the internet. That said here is my advice based on your description and response in the comments.

Congrats to your wife for being able to obtain entry into the US legally. Based on your description it appears that the immigration officer has stamped a No EOS/COS in your wife’s passport, this means that the CBP officer believes that your wife shouldn’t be allowed to extend here status or change your status.
The key here is that this is the opinion of the CBP agent and is not binding as the CBP agent isn’t the authority when it comes to deciding whether someone should be allowed to extend their status of to change it. This is exclusively the domain of USCIS.

Based on the USCIS website, you need to meet the following conditions to apply for an extension of status –

• You were lawfully admitted into the United States with a nonimmigrant
  visa
• Your nonimmigrant visa status remains valid
• You have not
  committed any crimes that make you ineligible for a visa
• You have not
  violated the conditions of your admission
• Your passport is valid and
  will remain valid for the duration of your stay

Assuming that these are true (Depends on if your wife’s passport is valid and if she committed any crime during her stay, if not the other conditions will be valid till the date the CBP officer stamped on your passport or on her i94), she is eligible to file for a request to extend her stay by timely filing a form I-539.

Now that we’ve evaluated the legality, lets also hypothetically discuss the next steps. Do note that this again is my opinion and not legal advice. Based on the facts, it is evident that your wife entered the US to give birth (while this in itself isn’t illegal) and to do so she lied to the CBP official. Given this when USCIS evaluates the application for extension it will most likely be denied, the only reason they would approve it it if your wife uncovered some information that was previously unknown to her; perhaps the baby is due prematurely or has a condition that she wasn’t aware of; you will need a US Doctors letter explaining such circumstances.

What does work in your wife’s favor is the time USCIS takes to process extension of stays and the fact that she will not accrue illegal presence until USCIS adjudicates her application. So if she was to file the paperwork timely and have the baby and leave the US before USCIS denies the petition she will have accrued no illegal presence.

This might still make it harder for her to seek admission to the US again nevertheless but she will not have done anything illegal strictly per the immigration law.

I wish you and your wife the very best and strongly urge you to consult an immigration attorney.

Your wife’s intentions are being confused.

They think she is here to give birth.

The US has had 243 years of continuous government. Most of its Constitution was written in the age of sailing ships and wagons for transport, when a pregnant woman coming to the US wasn’t really possible. This was just after the Civil War, which ended (state sponsored) slavery. To settle the question and lock in citizenship for former slaves, they made the Constitution say that citizenship is granted to anyone who was on US soil when they were born. This is called jus soli.

They might not have gone with that decision today, when that Peculiar Institution is a distant memory and a pregnant woman can easily, safely get to the US in 1-3 days. The only reason it’s still that way is that the Constitution is really hard to change. But the recent controversy may cause it.

Anyway, because of this old rule, many pregnant women enter the country specifically so they can give birth on US soil. They want to give their child the ”gift” of US citizenship. And US taxes.

After the child turns 21, he can apply for citizenship for his parents and other family members under current law (which is more easily changed, and is likely to be changed given current fervor about immigration). This creates the appearance of birth parents seeking citizenship for themselves also.

I’m sure this has nothing to do with your reasons, and perhaps this is the first time you’re hearing of any of this. The upshot is:

Don’t even create the appearance of "birth tourism"

Her situation is problematic.

She claims that flying is perfectly fine at 32 weeks (evidenced by the fact that she willingly did it). But now she claims that flying is not ok at 34 weeks. There are a couple problems with that latter claim. First, it contradicts the fact that she told immigration that was her plan!!! This has "deception" written all over it.

Second, it contradicts the fact that she booked the return trip that way, and started the trip knowing what the timing would be. So she changed her mind after clearing immigration. That makes it look like that was her plan all along.

And third, if she genuinely did just now develop this belief that flight after 34 weeks in unsafe, then why doesn’t she move up her return date and leave immediately? Have the doctor squeeze her in first thing tomorrow morning, then be on a flight tomorrow afternoon… That would be most prudent, staying in a strange foreign country to use the most expensive healthcare in the world doesn’t make a lot of sense on its face.

All these create the appearance of an ulterior motive, and a rather obvious one. And yes, there is no presumption of innocence in immigration; immigration is not a right.

So anyway, Immigration saw this coming a mile away. She would have been turned away immediately. She was only let in because Immigration saw a great deal of evidence in her favor, that her trip had other legitimate reasons, that she would be a reliable visitor and would keep her commitment. She earned that trust, clearly. But if she now stays, it will break that trust, making the deception even more serious. On top of that, she will have overstayed.

Immigration law calls for a lifetime ban for deception. Immigration will often reconsider after 10 years, but that’s not required. Until the Constitution is changed, it will probably be Immigration’s practice to make life as difficult as possible. Your future citizenship would be impossible because of deception. A constitutional amendment could even void the child’s citizenship as the product of fraud. It would make a big mess.

Staying to have the baby is simply not an option. She needs to complete her travel plans ASAP.

She should leave and go and have the child in Nigeria. Do not risk it for several reasons.

1. Public Charge complications
2. Visa Revocation/voiding
3. Immediate Removal/Deportation Proceedings
4. Fraud/Misrepresentation

Under the new USCIS policy, most immigrants are put into removal proceedings immediately their extension of status is denied. In you wife’s case she has only two weeks. The change of status will not be acted on within two weeks, that’s a guarantee. Thus if it’s refused it means she will begin accruing illegal presence immediately.There are no appeal rights for B-2 nonimmigrants who are denied extensions of stay.

Do you have insurance to cover the childbirth? If you use public assistance, that’s a public charge. Her next USA visa application will be definitely refused. Even if you petition her for a green card, the possibility of public charge denial will follow her.

Immediately she accrues one day of illegal presence, her visa will automatically void even if its a 5 year visa wirh remaining validity. She will then have to apply for a new visa, with unknown results.

Finally she can be barred from reentry for misrepresentation because I am very sure she told the immigration officer she does not intend to give birth here. Well if now she does give birth they will say she lied. That comes with a permanent bar.

See this video of a Nigerian woman getting her USA visa revoked for attempting to board a flight to give birth in the USA, forward video to 41:35. Your wife was very lucky. Don’t push your luck.

Bottom line, don’t do it. You’re from Nigeria. It might be helpful for you to know that under the Trump administration people from Sub Saharan Africa are being put into deportation procedures at twice the rate of other aliens.

Just don’t do it.

References

Black Migrants Vulnerable For Deportation

Sub-Sahara African migrants have seen the highest spike in
deportations during the first nine months of the Trump presidency—more
than doubling from 909 to 2,184, according to the report.

Trump administration is revoking previously issued nonimmigrant visas at a record pace for a variety of reasons

Michelle Nicoll Gutierrez had often visited the United States legally
since her mother married a retired U.S. Foreign Service official years
ago and relocated from Mexico to Maryland. Nicoll Gutierrez’s toddler
son was born here after she became ill during an extended stay late in
her pregnancy.

She was returning on another trip to celebrate her 42nd birthday when
federal officials detained her at Bush Intercontinental Airport
Saturday and denied her entrance, revoking her tourist visa, and
barring her from returning for five years.

But they said federal officials focused unusually on her use of
Medicaid and state health insurance during the birth of her son in
December 2016 in Maryland.