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As Bjorn says in his answer, the set of binomial coefficient functions just isn't sufficient in general. However, plenty has been written about analogues of Mahler expansions, i.e. finding nice bases for various spaces of continuous functions, going back to Amice in the 1960's for finite extensions of ${\mathbb Q}_p$, as well as positive characteristic versions. This is all very nicely explained in Keith Conrad's The Digit Principle, J. Number Theory 84 (2000), no. 2, 230--257. arXiv version

S. Evrard has recently extended some of these results to cases with infinite residue field Normal bases of rings of continuous functions constructed with the $(q_n)$-digit principle. Acta Arith. 135 (2008), no. 3, 219--230.

Edited to add: probably of more relevance to your question though would be the theory of Mahler-type expansions developed in p-Adic Fourier Theory by Schneider and Teitelbaum.

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As Bjorn says in his answer, the set of binomial coefficient functions just isn't sufficient in general. However, plenty has been written about analogues of Mahler expansions, i.e. finding nice bases for various spaces of continuous functions, going back to Amice in the 1960's for finite extensions of ${\mathbb Q}_p$, as well as positive characteristic versions. This is all very nicely explained in Keith Conrad's The Digit Principle, J. Number Theory 84 (2000), no. 2, 230--257. arXiv version

S. Evrard has recently extended some of these results to cases with infinite residue field Normal bases of rings of continuous functions constructed with the $(q_n)$-digit principle. Acta Arith. 135 (2008), no. 3, 219--230.