Estimating a phase for a frequency band
a single complex number in the FFT result table is, in fact, a representation of some (small and determined by the sampling rate and length of signal) freq band

Not really. The DFT is first and foremost a mathematical procedure that simply allows representing a time domain sequence as a sum of complex exponentials. The physical interpretation of that operation depends a lot on the circumstances but in general it's NOT true that FFT coefficient at some frequency $\omega$ represents accurately what's happening physically at frequency $\omega$. They are related but the exact nature of the relationship depends a lot on the details.
The notable exception here are time sequences that are periodic with the FFT length, but periodicity implies infinite length which doesn't exist in the real world.

want to estimate a phase (single number) for a frequency band of 9-11 Hz.

That would require you to first define what a single number phase for a frequency band is. Typically the phase is different at each frequency, so typically a single number doesn't
work well.