The first factory?

I caught up with Michael Portillo's Great British Railway Journeys trip to Cromford and Derby last night, which I'd taped. I was anxious to see this episode as I am researching the early mills for my book The Children History Forgot, which focuses on child labour in the eighteenth and nineteenth centuries. I was very surprised to hear Mr Portillo say that Richard Arkwright's factory at Cromford was the first water-powered factory in Britain. There was an even earlier silk mill built by George Sorocold for Thomas Cotchett at Derby in 1702, which had a waterwheel, but was not a success. (People and Industries, W.H. Chaloner, Frank Cass, 1963.)   He then contradicted what he'd said earlier by saying the first factory was built in Derby - which is correct!  Although Arkwright's was the first successful COTTON factory, the silk mill at Derby built by the Lombe family circa 1721 was the first viable water-powered textile factory in England.  If I remember rightly, the Old Silk Mill behind Mr Portillo in his piece to camera was built on the site of the Lombe mill. I was also surprised that Mr Portillo trotted out the hoary old chestnut about John Lombe being poisoned by an Italian woman for bringing the secrets of powered silk throwing to England, because this story was discredited by historians years ago. Makes good telly, I guess. Mr Portillo's visit to John Smedley's factory was lovely to watch, though - inspiring to see we still have some British industry left. We have lost so much.
Image: Sir Thomas Lombe's silk mill, Derby, completed in 1721. Charles Knight's Pictorial Gallery of Arts, Vol.1, c.1860.