The paper gives conditions for a submersion \(f:M\to N\) between paracompact Banach manifolds (with \(N\) connected) to be a fiber bundle. A direct connection is made between the fiber bundle structure and suitable path-lifting properties, providing criteria of topological nature (such as \(f\) being a proper or a closed map or of metric nature). In particular, in the case of Finsler manifolds in the sense of \textit{R. S. Palais} [Topology 5, 115--132 (1966; Zbl 0143.35203)], ``Lusternik-Schnirelman theory on Banach manifolds'', assuming \(M\) complete, metric requirements such as the Hadamard integral condition are provided. For the case when \(f\) is a local diffeomorphism, conditions are given for \(f\) to be a covering or a global diffeomorphism. The results are adapted as well to submersions with uniformly split kernels in the sense of \textit{P. J. Rabier} [Ann. Math. (2) 146, 647--691 (1997; Zbl 0919.58003)].