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I am no expert in these matters, but I was just consulting the Collins Butterfly Guide and it says that male Essex Skipper has a broken sex brand at v2. I can see a break in the sex brand quite clearly on this picture. The underside of the antennal tips are not obvious in the picture (the other separating feature between Essex and Small Skipper). Also the distribution map in the guide suggests that Eastbourne would be well within the range of Essex Skipper, but I don't claim any knowledge of the Eastbourne area.
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Then again, looking at photos of the two species online, the sex brand looks too long for Essex, and some small skippers seem to have a slight break in about the same area, rather than being a continuous line as suggested in the guide. Also in the photos of Essex, the wing veins on the fore-wing are less black/absent until the submargin/margin of the wing. All part of the learning process I guess.
[Edit] UK Butterflies has a very good description of the difference in sex brand at the bottom of the Essex Skipper and Small Skipper pages. See: http://www.ukbutterflies.co.uk/species.php?species=lineola
Thanks for the link. It could be either...
Having looked at it in more detail I'd be more inclined to say Small Skipper, but you are probably right to er on the side of caution.
Lat/Lng: 50.7455, 0.262347
OS grid ref: TV597964