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This is a cross post of MSE post somehow:

Is there any example of compact fiber bundle $E$ with noncompact fibers $F$?

Obviously if the base space $B$ is $T_1$ then there is no such example.

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    $\begingroup$ Hint: use compactness of the base $B$ to show that $E$ is trivialised over some nonempty closed subset $Z \subseteq B$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 3, 2021 at 21:18

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