Friction shifting 9 speeds using an 11 speed rear derailleur
It won't necessarily just work, and there is a way of predicting whether it will with good accuracy, barring a borderline case.

If it's a Shimano 7, 8, 9, or 10 shifter in friction mode, it probably won't work.

Friction shifters have some variance in their total cable pull between models. It's easy to measure by marking the cable and using a ruler or caliper to note the total movement of the mark. 

Take that total cable pull number and multiply it by the actuation ratio of the rear derailleur. For Shimano road 11, it's basically 1.4. That number has to be equal to or greater than the total cog-to-cog distance of the cassette, which for Shimano 9 is about 34.8mm. If it's less, the rear derailleur won't have enough total movement.

The reason there can be borderline cases is because rear derailleur movement for a given input is actually a curve, not a perfect ratio, and the exact start to finish value is going to depend on the high limit setting. But it's pretty close to just being a ratio.

A 7/8/9/10 Shimano shifter in friction mode has a little bit of movement beyond where its last detent position would have been, but not very much. Since these generations are designed to pair with a 1.7 actuation ratio rear derailleur, that small bit of movement is not likely to be enough to compensate for the fairly large difference in actuation ratio.

I don't have any experience putting 9-speed chains in 11-speed derailleurs. Sometimes that direction of mismatch can cause rub between the derailleur inner cage and the chain. 9 speed chains don't tend to experience this in 10 speed derailleurs and 11 speed chains are only something like 0.2mm narrower than 10, so 0.1mm per side. It will probably be fine to run 9 but you could do 10 or 11 to be safe. The shifting may be laggier but it's unlikely to matter with friction.