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On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some of the features of a functional derivative and the covariant derivative."

My question is what does this derivative look like, explicity (for a Finsler manifold over a general Banach space)?

Can someone provide a reference also, much appreciated.

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    $\begingroup$ a nonlinear connection that defines the covariant derivative in Finsler geometry is here --- is this what you want? $\endgroup$
    – Carlo Beenakker
    Commented Nov 4, 2015 at 5:23

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Chapter 2 in Bao, Chern, Shen: An Introduction to Riemann-Finsler Geometry may give you the answer.

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