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Drift chambers operated with helium-based gas mixtures represent a common solution for tracking charged particles keeping the material budget in the sensitive volume to a minimum. The drawback of this solution is the worsening of the spatial resolution due to primary ionisation fluctuations, which is a limiting factor for high granularity drift chambers like the MEG II tracker. We report on the measurements performed on three different prototypes of the MEG II drift chamber aimed at determining the achievable single-hit resolution. The prototypes were operated with helium/isobutane gas mixtures and exposed to cosmic rays, electron beams and radioactive sources. Direct measurements of the single hit resolution performed with an external tracker returned a value of 110 $\mu$m, consistent with the values obtained with indirect measurements performed with the other prototypes.