Hello,
You can have small spiral antennas, but the relative bandwidth (BW/CenterFreq) will not be large anymore, or you have to sacrifice radiation efficiency, gain.
Most realy wide band antennas have their lowest usable frequency dictated by half wave or quarter wave electrical length. At their highest frequency, their size is many wave length. Wideband spiral antennas are related to traveling wave antennas.
I suppose there is a relation with your posting on bandwidth ratio?