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Yes. Consider any local system (EDIT: of rank 1) over a characteristic 0 field on $\mathbb{C}^*$ with non-trivial monodromy. This satisfies all of (1), (2), (3) and (4). There are lots of ways to check that this has trivial cohomology; for example, if the monodromy has finite order, it's a summand of the pushforward from the constant sheaf, which has the same cohomology as the constant sheaf. If you want a projective example, an elliptic curve works.