Interesting subject. Not only for me but it has many views on this forum; I'll add my 2 cents.
There is a chance of some harm on negligent conditions.
I am not saying permanent blindndess. And thoose negligent conditions can be precisly using a powerfull flash in close range.
I am not an expert. And reading the answers on this posts, and the references quoted and linked, it seems that there is no real expert on the particular subject out there. (I'll explain the statement "no real expert")
Some links come on a discussion forum. Yes answered by ophthalmologists, but using the words:
"is quite unlikely"
- "the light from a flash is too unfocused and of low intensity" Which is circumstantial, becouse it can be very focused and inclusive for that fraction of a second can overpower Sun's light.
There is no real expert becouse no one will conduct a methodic experiment firing flashes to infants.
Flash blindness is a well known effect in military aplications. This article says that some sources say it can be temporal or permanent (the original link to the defense department is broken): https://en.wikipedia.org/wiki/Flash_blindness but also says
"temporary flash blindness" when discussing everyday flash photography emphasizes that the condition will disappear without ill effect.
So there is a condition. The point is that if it is permanent damage or not.
To make a methodical study it should be one that includes specific waveleinght's, duration, intensity, specific damage (burn on specific parts of the cell, or chemical unbalance of the receptors), duration of this effects, celular regeneration of the afected area, pupil's aperture on the moment of the exposure, etc. I do not know such a study.
Common mistake is that only UV light can hurt tissues. But a burn is not made only by UV light, but can be also becouse visible light and infrared light (among others). A light of the sun focused by a magnifier. The cristalline is a focus lens. Much smaller aperture than a magnifier but is one.
Can an empirical statistic study be this one?
"we would by now have an epidemic of damaged eyes"
I don't think so. But it helps.
If it is not a 100.000000% sure, there is a chance eyes (not only babies) hurted in some conditions, negligent conditions if you like.
So... Do take precautions
You do not want to be the exception to the statistics.
So take your precautions as everyone has mentioned. This also aplies to portrait photography.
Bounce the light.
Use a slight higher iso, like Iso 200.
Do not use f/32... Use a wider aperture, f/1x-f/8 perhaphs?
Turn on some abmient light too to help the pulil reduce its aperture a bit. If you are using a softbox on a studio flash turn the model light on.
Some aditional notes
This does not only applies to babies but to portrait photography. You do not want your model or client to feel too unconfortable with this temporary flash blindness. Talk to them and do not make your studio dark.
The flash easily overpowers the ambient light of a studio, so including the case of low key photography you do not need to be in darkness.
But on normal conditions do not worry
As this is a photography forum, any normal situation you encounter will not damage the eyes.
The article @Irme posted has an interesting summary.