Does real life have "update lag" for mirrors? This may sound like a ridiculous question, but it struck me as something that might be the case.
Suppose that you have a gigantic mirror mounted at a huge stadium. In front, there's a bunch of people facing the mirror, with a long distance between them and the mirror.
Behind them, there is a man making moves for them to follow by looking at him through the mirror.
Will they see his movements exactly when he makes them, just as if they had been simply facing him, or will there be some amount of "optical lag"?
 A: When I read a few answers to this question, I noted how everyone wants to emphasize that it is a small effect because light moves so quickly. So that set me pondering whether we can find an example with longer distances.
If we look out into the depths of space, we don't see mirrors but we do see something that acts to direct the path of light, namely gravitational 'lenses'. A gravitational lens is just a heavy thing such as a galaxy that happens to be on or near the line of sight from Earth to some other galaxy. It can then happen that light can travel from the further galaxy to Earth by more than one path, and the paths do not have to be of the same length. So we end up seeing the same distant galaxy at two or more different moments in its life, all at the same time (i.e. time of arrival of the light) for us. And the time differences now can be long: not nanosecond or seconds, but years or millennia. Any friendly astronomer reading this may care to give examples.
A: As the speed of light is finite, sure enough there is some lag, but let's evaluate how big that lag is. Considering that the mirror is 100 meters away, than the lag will be $$2\times 100\: \mathrm{m}/(3\times 10^8\:\mathrm{m/s}) = 667\:\mathrm{ns}.$$ Comparing it to average human reaction time of about $0.1\:\mathrm{s}$, one can conclude that it is impossible for a naked eye to notice any lag at all.
A: If there were such a lag, you could use a moving mirror to measure the speed of light.
This was first done in the 1920s by Michelson. (See e.g. this link which hosts papers from 1924 and 1927.) Michelson built an octagonal solid with mirrored sides which spun at several hundred times per second, and used this spinning mirror to measure that light took about 230 microseconds to make the 44-mile round trip between Mt. Wilson and another mountain, whose location is not uncertain but whose name is different in different reports.
Human persistence of vision is measured in milliseconds, not microseconds, and the unaided human eye can’t resolve images in mirrors from such distances. So this isn’t an effect that you could observe naked-eye in a mirror that fit in a stadium, for any reasonable definition of “stadium.” The lag is real, but it takes experimental finesse to observe it.
An astronomical version of “mirror lag” is a supernova light echo.
A: This is a simple distance problem the distance from your eye to the mirror and the distance of the man from the mirror added together.  The size of the mirror makes NO difference at all in speed of light but some in how far the object is away and the prior answer was correct except he was using the speed of light in a vacuum which you are not in.
Now, the most important condition is that you are able see the man's gestures in the mirror. I would say that he would have to be no further than around 1 mile in total (his and your distance from mirror added together) for an average person to be able to see his gestures. So the formula would be 1 mile multiplied by the speed of light in atmosphere. Don't forget elevation; it affects air density.
A: Even without a mirror there's a lag, because it takes time for light to go from the man to the audience. Divide the distance from the man to the audience by the speed of light to get the delay.
With the mirror, the light has to flow from the man to the mirror and then reflect to the audience members. Therefore, the time between the man making the moves and the audience seeing them is essentially the same as if his distance from the audience were the sum of his distance from the mirror and the audience's distance from the mirrow. If he's right behind the audience, going through the mirror doubles the lag.
The process of the mirror reflecting the light will also take some tiny amount of time, but I don't know how to quantify this. I suspect it's negligible compared to the differences in distance for different audience members.
And as others have pointed out, the speed of light is so fast compared to the reaction time of the human vision system that all these times are essentially instantaneous as far as our perception is concerned. But we can create devices that can detect these delays (RADAR and LIDAR work by measuring the time it takes for a radio or light transmission to reflect off something).
A: There will be an optical lag due to the finite speed of light, $c$. For light to travel between two points separated by distance $d$ the lag time is
$$
t = d \times \frac{1}{c}
$$
The speed of light is
$$
c\approx 3\times 10^8 \text{ m/s} = 30 \text{ cm/ns}
$$
$$
\frac{1}{c}\approx 3.3 \text{ ns/m}
$$
So, in words, light travels $30 \text{ cm}$ in $1 \text{ ns}$, and, equivalently, it takes light $3.3 \text{ ns}$ to travel $1 \text{ m}$.
I leave it as an exercise for the reader (or commenter) to express $c$ and $1/c$ in "convenient" imperial units.
A: Yes, there is a lag-time.
The speed of light is 299,792,458 meters per second.
Suppose we have you, a one-meter wide mirror, and a baseball.
You and the mirror are 20 meters apart from each other.
The mirror is not moving relative to you.
The baseball is traveling one meter per second slower than the speed of light, relative to the mirror
As you stare into the mirror, the baseball flies by.
You would not see the baseball in the mirror until a fraction of a second after the baseball has already finished passing the mirror
