Paul Hawker, Station Manager at Buxton Fire Station, says people battled in vain to save the driver whose car plunged into the icy river.
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The road is expected to remain closed for several hours, and police are advising motorists to take alternative routes where possible.
More than 800 Shakespeare scholars from over 48 countries are gathering in Stratford Upon-Avon and London.
It comes as figures show there has been a rise in trespassing in the last three years across the West Midlands.