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Olivier Benoist
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Here is a counterexample if the map is not proper. Consider the inclusion $f$ of the plane minus a point $U$ in the plane $X$. The line bundle $\mathcal{O}_U$ is ample restricted to the fibers of $f$ (they're points...). However, it is not $f$-ample. Indeed, if it were, $\mathcal{O}_U$ would be ample on $U$. Choosing $N>>0$, we would get $$H^1(U,\mathcal{O}_U)=H^1(U,\mathcal{O}_U^N)=0.$$

But a simple computation via Cech cohomology shows that this cohomology group is not trivial (in fact, infinite).

Olivier Benoist
  • 6.5k
  • 2
  • 38
  • 55