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Many years ago I got visited relatives in Potters Bar and they collected us from Cockfosters station...
On the journey they explained how the Picc line should have gone on to Potters Bar but most the land was owned by the Crown and they wouldn't allow the track to go on the surface and it was too expensive to tunnel the 2.5 miles...
Therefore the line ended at Cockfosters

Is this a "Cock and Bull" story an urban myth or factual

I can find no evidence on various internet searches.....

Until evidence is found to the contrary, it must be considered a myth.

The myth could have been perpetrated due to the original northern terminus of the Piccadilly line being at Finsbury Park. There was a connection with the Great Eastern Railway at that point, and so trains could, in theory, have been run out on to the main line.

Having done my own investigations, I can find nothing that says "Potters Bar" on it with regard to an extension. The closest thing I could find was some crappy "Answers"-type website, where someone asks if the Piccadilly Line will ever be extended to Potters Bar. It'a a weird question, and has no context.

I would say that if the idea had been mooted and seriously considered, then we would be able to find something about it.

Yes, I mean you can find a lot of info about the proposed Northern Line extention to Bushey Heath, I know it wouldn't have been at the same level of planning, but there would be something on the web if it was true.

I've just checked John Glover's book London's Underground which is generally comprehensive and has details of many other abortive extension schemes, but no mention of a northerley extension from Cockfosters.