I wouldn’t use the phrase for a body of water: There, if it was flat enough to require a simile for emphasis, it’d be “smooth as glass”, or the like. Something relatively smooth but short of that standard, though, would just be “very smooth”.
Kansas, though, yeah, that’s flat as a pancake.
No discussion of flatness of pancakes can be complete without a cite to the Annals of Improbable Research, the folks who award the IgNoble prizes.
Back in Ye Olden Dayes of Yore (2003) they published a scholarly article “Kansas is Flatter than a Pancake” which proceeded to prove the same in the flatness not thinness sense; Kansas having much less vertical relief on its scale than does the standard American pancakes the scientists examined.
Sadly it appears internal link rot has taken over their own site and the paper is not directly available. But here’s a page of their articles about their ongoing articles about flatness, pancakes, and Kansas.
Each of the articles on that page has some good humor worth taking in. And after all, ice cream is good on pancakes too. 
I think that, with respect to the Milky Way galaxy, both types of pancake are correct. The Galactic Thin Disk is as flat as an English pancake, with a ratio of around 100/1 (diameter/depth). However, the Galactic Thick Disk is more like an American pancake.
That diagram does not seem to comport with @Chronos comment that the proportionate thickness of the galaxy is comparable to a CD - even the thin disk looks quite a bit thicker than a CD to me? And indeed tracing down the image it is annotated as
not to scale
I’m not saying the thickness proportion shown there is necessarily wrong, it may be only some other aspect of the diagram that is not to scale. I’m not exactly sure how to apply the concept of “scale distance” here. Is it reasonable to use the scale height : scale length ratio to determine what we’d informally think of as the proportionate thickness?
The Thin Disk is a bit more than 1000 ly thick and 100,000 ly wide. The Thick Disk is much thicker (and about the same width), although the boundaries are less well-defined. I included the diagram to show the location of these structures, rather than their extent.
Ok, so 1:100 is indeed more CD-like than what’s shown in the diagram, right?
[ ETA: Precisely so, in fact - I just looked it up, a CD is also exactly 1:100.]
Well, it probably isn’t reasonable, because we are talking about nebulae here, and they are nebulous by definition.
During the Apollo era it amused me when I figured out that, however temporarily, the astronauts sitting at the top of their Saturn V were the highest point in the state.
Cape Canaveral being under federal jurisdiction, however, I think we can stipulate that they were not examples of “Florida Man”.
Well, in America, that would spark a debate as to whether thin crust actually qualifies as Pizza. Expect the Second Coming of The Great American Civil War in the fight between New York Style and Chicago Style.
Now the highest point in Florida is a skyscraper in Miami.
Nebulae, and galaxies (not the same thing), are both nebulous, and without sharply-defined boundaries, but the scale height is a sort of “average thickness” (of course, it’s more rigorously mathematically defined than that), so it’s about the best you can use for a measurement.
We had a similar discussion not long ago. I, as a native Kansan, argued that Kansas is by no means as flat as a pancake. With a high elevation of 4,039 feet near its Western border, and a low elevation of 679 feet on its Eastern border, Kansas is more akin to a ramp than a pancake.
That’s because God is tipping the frypan while cooking the Kansas pancake.
And in a few weeks it might be a different skyscraper in Miami.
IANA expert, but Saskatchewan seems to be simply a larger & colder Kansas. A few hills at the western border, otherwise real flat, but continuously sloping downwards to the east.
I’ve not seen any comparative studies. 
Yeah. I looked into what was the highest in Miami a while ago and it seemed like a new tallest was being built every couple-three years. So I didn’t bother to memorize which one was highest at that point.
Now if these guys are smart, they’d build the bottom few floors to be able to hold up when under salt water and put boat docks on the second and third floors. No sense in having their investment become worthless because some ice melts.
never mind
I didn’t mean this, although from the second half of your comment maybe you did understand my tasteless attempt at humor? Given recent events, I was implying that in a few weeks the current second-tallest building in Miami might become the tallest.