Perhaps it is. As 'cane it' also means 'to overindulge in drugs or alcohol' in the UK, I presumed it had something to do with cocaine. That seemed logical (in my mind) as being to do with going faster, but it might not be anything to do with that at all. As you can also go 'on the lash' meaning to get drunk, I suppose that both lash and caning might be something to do with harming yourself. On the morning after you might agree that you've given yourself a beating. Only speculation, of course, and I think I'm going off on a tangent here.To "cane" someone means to "whip" them with a stick.
To make a race horse run faster you would whip the horse.
Perhaps this is an extended metaphor.