Suppose $(E,p,B;F)$ is a fiber bundle such that $E$ is homeomorphic to $B\times F$, is it true that the fiber bundle is trivial?

Suppose $(E,p,B;F)$ is a fiber bundle such that $E$ is homeomorphic to $B\times F$, is it true that the fiber bundle is trivial? A non connected counter example has been provided, so I'll ask for E,B and F to be connected (hopefully low dimensional) manifolds.

Suppose $(E,p,B;F)$ is a fiber space such that $E$ is homeomorphic to $B\times F$, is it true that the fiber bundle is trivial?

Suppose $(E,p,B;F)$ is a fiber bundle such that $E$ is homeomorphic to $B\times F$, is it true that the fiber bundle is trivial?