No you can't. The BBC would charge you and if course you would be breaking their copyright.
ITN offer a buyout/rights management option on archive footage. However, clips aren't cheap. I recently needed a 70 year VT clip of a sussex town for a documentary I am working on. The cost? �149 for 35 seconds.
As for the BBC, their BBC Motion Gallery is the best place to go.
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Do you know the answer?
Is it free to use BBC material, such as news reel and archive footage, in your own commercial film production?