MentalFloss:
Edit: I Misinterpreted the current gain. It's not expressed for Ib but for Ic. Basically 5mA will do fine on the base, but 22mA is no problem either
Current gain isn't relevant here, you are using the transistor as a switch so you
want the section of the datasheet showing Vsat against base drive for various
levels of collector current.
gain is for the linear region (collector > emitter voltage + 2V or more)
Vsat (saturation voltage) is what determines the performance as a switch
and you want it as low as practicable to reduce heat-dissipation as you
seem to have discovered. For 800mA load try 60mA drive (ie more than an
Arduino can supply)
Unless you go for a modern super-beta device you'll find base drive is normally
0.1 x collector current for switching transistors.