Since nothing exactly like this exists on Mars, let's look at the best real-life example of what you're describing: A crater.
Hellas Planitia is one of the oldest and largest craters on Mars. It was formed during the Late Heavy Bombardment, which started around 4.1 BYA. At more than 7 km deep, it's one of the deepest craters in the solar system.
This crater has survived nearly all of Mars' volcanic history, atmospheric conditions, erosion from wind and water, has even been dumped into by a few volcanoes, and is still this large after billions of years. Assuming you maintain a similar grade (maybe even steeper depending on your timeframe), your pit could theoretically last as long as the history of the planet.
Since you also mentioned atmospheric pressure, it's worth noting that at the bottom of the crater the atmosphere is already 103% more dense than at the surface of the surrounding topology.
Edit based on comments: Since nothing of the dimensions described exists in the solar system, let's check out some examples of rock slopes here on earth.
Trango Towers in Pakistan is home to some of the steepest rock faces in the world. They range up to more then 7 km tall, and have nearly-vertical drops, so the downward pressure of gravity is much more significant than the outward pressure of the rock. That being said, you're talking about something below the surface, so let's look at perhaps a closer example.
The Mariana Trench dips down to 11 km below sea level, over a third of the value we're targeting, so it should be a decent model for how these things work on a large scale. According to studies of the trench, the deepest parts still maintain an incline of up to 34 degrees in places. Consider that the trench is entirely submerged, is subjected to massive erosive currents, and violent earthquakes, and it has survived for 180 million years and counting.
As long as you stick to a gradient of ~30 degrees (to be very very safe), I think any depth that does not break the crust would be fine. Just be careful to also consider temperature at that depth, as you'll be getting close to the mantle.