The type of whip is of course a cat o' nine tails.
Wiktionary disputes the theory that the saying refers to this instrument of punishment on the grounds that the first recorded reference to the cat o' nine tails came 30 years later than that of the phrase "not enough room ... " (1695, as compared with 1665). It's perfectly possible, it seems to me, that both idioms had been in use orally for some time before they were written down, which would make it nigh–on impossible to say which came first. I am extremely sceptical regarding the idea favoured by Wiktionary – that the "not enough room ... " saying is a reference to the actual swinging of a feline quadriped.
© Haydn Thompson 2017