Performing DFT of streaming audio problem. Is there a limit?
The window size of your FFT limits your resolution and there is no way round this (i.e. you can't zero pad or zero stuff to get round it). The easiest way to think about it is: lower frequencies have longer periods so need a longer window (more samples) to differentiate.

I believe it would be impossible to increase your resolution even if you used an alternaitve method like a series of very steep band passes narrower than the frequency resolution permitted by FFT for your window size. Yes, you would get some measures out from your band passes but if the window size was not long enough, the data would be highly inaccurate in differentiating very close lower frequencies as there would not be enough data to disentangle them.

Consider a window of 1 sample or 2 samples, in these cases it is easy to see that you just can't recover resolution from small windows without making them bigger by gathering more independent samples.