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Are there complete fields of positive characteristic with non-trivial absolute value? What does calculus on them looks like? I'm aware that they have to be non-archimedean, and that the bulk of results about power series convergence will carry over, but surely there have to be some extremely weird results e.g. local expansions will be ambiguous.

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The field of quotients of the ring of formal power series over a finite field is an example of such a field (the link given by @MarcvanLeeuwen seems to be about characteristic zero only). –  Jyrki Lahtonen May 22 '12 at 9:57
@JyrkiLahtonen Thanks for that precision; I had been a bit too quick in posting that. I do think that the study of such fields as you mention are generally considered to be part of $p$-adic analysis. –  Marc van Leeuwen May 22 '12 at 10:04
@JyrkiLahtonen You mean $k[[X]]$ with the usual non-archimedean absolute value? Where can I read about calculus on this thing? –  Alexei Averchenko May 22 '12 at 10:11

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