Have one on my desktop and although I completed the download and ended up with a .zip file, the .part.zip one remains. Switching the PC off changed nothing. When I try to delete it says the file is being used by another programme so it cannot be deleted. Anybody knows?

what were you using to download it? open task manager and see if something related to that is running and kill it if so

Else use ProcessMon to search for the filename in the handles section and find out for sure which process is holding it open

this might help
http://ccollomb.free.fr/unlocker/

EDIT: In fact it will definately work. I have it installed on my PC for such things. It works i nthe context menu (right hand click)

My brother was sending me a zip file via Skype. I received it no problem.
Process Monitor seems beyond me, ccollomb is not free. Any othr ideas please?

You could try deleting in safe mode or killbox would do the job.

unlocker has done the job

I was going to say that unlocker is free – not sure how you thought it wasnt…
Its handy bit of kit.