Pitch detection algorithm for minsample size
Consider trying an upsampled or interpolated ASDF, AMDF, autocorrelation or other similar periodicity estimation algorithm.

There in an information theoretic time versus frequency resolution versus noise trade-off.  At a sample rate of 44100, estimating 440 Hz +-2 Hz might require somewhere in the range of 2 to 6 times 44100/440 samples (to determine the existance of periodicity or pitch) up to 2.5 to 4 times 44100/2 samples (to separate 2 spectral lobes), depending on the S/N ratio, and whether that noise environment requires windowing, or allows sub-sample or sub-bin interpolation in your periodicity estimator and/or FFT.

BTW, "super resolution" for tuners is usually considered to be +-1 cent or better, even though this is below normal human audibility except for beating.