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Has a Serre fibration $f:E\to B$ with $B$ a connected space isomorphic fibers over different points of $B$?

If $f$ is a fiber bundle, then all fibers are isomorphic. Hence, a possible counterexample would be a Serre fibration which is not a fiber bundle. It would be nice, if someone could provide a counterexample where $E$ and $B$ are non-pathological spaces, e.g. CW complexes.

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    Yes, I mean homeomorphic, isomorphic in the category of topological spaces.2011-12-02

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No, not always. In a counterexample below most fibers are intervals, but the fiber over one point is a point.

Fibration

Fibers are always homotopy equivalent, though.