Depending on your definition of "extended," this already happens with the moon. Consider also that Saturn's rings do cast a shadow on the planet. In any case, nothing in orbital mechanics prohibits this!
As a side note, I doubt a cloud of debris would have the same effect as a solid body like the moon unless it was REALLY BIG. Smaller debris clouds would dim light shining through, but the light would still go through at random points. There is also the issue of multiple reflections within the debris cloud, which allows more light through than a solid body.
How to Solve This
In theory, any orbiting body of sufficient size could extend nighttime with the right size, distance, and speed.
Specifically, if you wanted exact numbers, you need an angular velocity of the orbiting body relative to the perspective of the planet to extend this darkness time. Once you have this (and how long it extends night by), you can then solve for mass and distance of the orbiting body.
This will give a family of solutions, actually, because of the combined effects of mass and distance. So you will need to get an equation for the apparent size of the orbiting body, and fitler down your family of solutions by parameters that fit. To an observer on the surface of the planet, full eclipses are only possible when the apparent size of the sun is equal to the body providing the eclipse.
You may still have multiple solutions here, so this is where you get to choose a mass or size and solve for that.
A large enough sun-shield in a particular orbit would permanently shade a spot on the planet: specifically a Lagrange point of the planet-sun system. Go a little higher or lower in that orbit and this dark spot will move (more or less slowly, depending on the orbit).
Another option is a moon in a highly elliptical orbit: it will spend a lot of time in the outer range of its orbit, potentially causing longer nights/eclipses. This adds a little complexity to our approach by varying the apparent size of the moon, as it looks smaller the further it is from the planet and still needs to dim/block the sun. There is a much smaller "sweet spot" for this situation than if we decided to use mostly circular orbits.