Borel’s theorem for C ∞ -functions on a non-archimedean valued field
Compositio Mathematica, Tome 55 (1985) no. 3, pp. 289-294. 
@article{CM_1985__55_3_289_0,
     author = {Schikhof, W. H.},
     title = {Borel{\textquoteright}s theorem for $C^\infty $-functions on a non-archimedean valued field},
     journal = {Compositio Mathematica},
     pages = {289--294},
     publisher = {Martinus Nijhoff Publishers},
     volume = {55},
     number = {3},
     year = {1985},
     zbl = {0579.26007},
     language = {en},
     url = {http://www.numdam.org/item/CM_1985__55_3_289_0/}
}
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Schikhof, W. H. Borel’s theorem for $C^\infty $-functions on a non-archimedean valued field. Compositio Mathematica, Tome 55 (1985) no. 3, pp. 289-294. http://www.numdam.org/item/CM_1985__55_3_289_0/

[1] D. Barsky: Fonctions k-lipschitziennes sur un anneau local et polynômes à valeurs entières. Bull. Soc. Math. Fr. 101, (1973) 397-411. | Numdam | MR | Zbl

[2] W.H. Schikhof: Non-archimedean calculus (Lecture notes). Report 7812, Mathematisch Instituut, Katholieke Universiteit, Nijmegen (1978). | MR | Zbl