Air is 1% argon. Argon is heavier than air.
Why doesn't the argon concentrate in low-lying areas, choking out life there?
It does. You would find the average percentage of the atmosphere that is argon is very slightly higher at the floor of valleys. However, bear in mind first of all it wouldn't be anywhere near a complete stratification -- a layer of pure argon, then another of pure N2, and so on. A mixture of nearly ideal gases doesn't do that, at least at equilibrium, because it would eliminate the considerable entropy of mixing. (It can happen in liquids because liquids have strong intermolecular forces that normally favor separation and oppose the entropy of mixing.) Another way to think about it is that since the atoms and molecules in gases don't (much) interact, there's nothing stopping an individual argon atom going slightly faster than nearby nitrogen and oxygen molecules from bouncing up higher than they do.
What you would get in a theoretical ideal (uniform gravitational field, complete stillness -- no wind -- and uniform temperature) would be an exponential fall of pressure with altitude, and the exponential for heavier gases would be steeper than for lighter gases. That would result in enrichment of the heavier gases at lower altitudes. A little work starting from the Boltzmann distribution of gravitational potential energies of each type of atom and molecule would get you an ideal estimate of the argon excess as a function of altitude.
In practice the lower atmosphere has so much mixing due to wind and big thermal gradients that I doubt you could even measure the mild excess of argon and other heavy gases.
There is one fascinating short-term exception, which bears directly on your question. Sometimes volcanoes will belch out a considerable quantity of CO2, which is significantly denser than air, and this CO2 can accumulate briefly in a thick layer at the bottom of a valley or over a lake, if there isn't much wind. It can persist for some hours, perhaps days, before it diffuses away and is mixed with the rest of the atmosphere.
Then indeed the valley bottom becomes an invisible death trap for humans and animals: walk into the valley, or be unable to exit fast enough when it happens, and you will suffocate for no reason you can see. The most famous example of this is the Lake Nyos disaster in 1986 which killed thousands of humans and animals. I think the government now has mixing devices installed in that lake to prevent any future sudden release of CO2.