Geographically, Lincoln could be considered a derby. However, it has never had the intensity of our main derbies: Chesterfield or Notts County. Lincoln sit in that group of clubs who aren't too far away, but it is debatable as to whether it is 'derby' or not (along with the likes of Doncaster, Grimsby, Rotherham etc.).
Such is our intensity with Chesterfield, that is is difficult to feel the same way for any of the other localish clubs - In the same way between Arsenal & Tottenham, yet they don't have the same feelings towards their other London neighbours.
Rivalry is an interesting concept. For example, when we were in the same league as Alfreton it was difficult to have any strong feelings against them, as we had never been 'rivals' before - and we never saw them as a 'threat', so we didn't view them as rivals as much as clubs like Luton or York.
It is only recently that we are starting to become rivals with Lincoln in that we are both on the up, have similar ambitions and are competing for promotion. And the pot has been further stirred recently because of all the ex-Stags players at Lincoln.