Geodesic
On the Cauchy Problem for Differential Equations in a~Banach Space over the Field of $p$-Adic Numbers
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 99-106

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For an operator-differential equation of the form $y^{(m)}(z) = Ay(z)$, where $A$ is a closed linear operator on a Banach space over the field of $p$-adic numbers, conditions on the initial data are given that are necessary and sufficient for the Cauchy problem to be well-posed in the class of locally analytic vector-valued functions. The result is illustrated by $p$-adic partial differential equations. 
@article{TM_2004_245_a10,
     author = {M. L. Gorbachuk and V. I. Gorbachuk},
     title = {On the {Cauchy} {Problem} for {Differential} {Equations} in {a~Banach} {Space} over the {Field} of $p${-Adic} {Numbers}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {99--106},
     publisher = {mathdoc},
     volume = {245},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2004_245_a10/}
}
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M. L. Gorbachuk; V. I. Gorbachuk. On the Cauchy Problem for Differential Equations in a~Banach Space over the Field of $p$-Adic Numbers. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 99-106. http://geodesic.mathdoc.fr/item/TM_2004_245_a10/