Following up the intriguing finding by H. M. Rodrigues and J. Solà-Morales [J. Differ. Equations 269, No. 2, 1349–1359 (2020; Zbl 1451.37033)] of a continuously Fréchet differentiable self-map on an infinite-dimensional separable Hilbert space having the origin as a fixed point which is exponentially asymptotically stable but not linearly stable, this paper articulates more precisely the connection between linear stability (LS), exponential stability (ES), and exponential asymptotic stability (EAS) in a more general environment: a Banach space. In such a space, it is shown that LS and ES are equivalent, and that ES implies EAS. Moreover, the converse of the latter is shown to be true if the map satisfies a certain spectral gap condition which holds in the case of finite-dimensional spaces, confirming the absolute necessity of infinite dimensionality in the above finding.