I'd like to write a better response, but I must be brief.
For now, let me offer some places to read. Long story short, it is predicted that there's a relationship between special values of $p$-adic $L$-functions and syntomic regulators (which are the analogue of Beilinson's regulators in the $p$-adic world).
The beautiful paper of Manfred Kolster and Thong Nguyen Quong Do is, I think, a very readable resource.
The best results I know in this direction are Besser's papers here and here, which use rigid syntomic cohomology.
Besser's overview talk at the conference in Loen (notes available here) was a real joy.
It is false for the valuation ring in any nontrivial finite extension of $\mathbb{Q}_p$. The coefficients of the Mahler expansion of a continuous function $\mathcal{O} \to \mathbb{C}_p$ are determined by its restriction to $\mathbb{Z}_p$ (they are given as $n$-th differences of the sequence of values on nonnegative integers, in fact). But there are different continuous functions $\mathcal{O} \to \mathbb{C}_p$ with the same restriction to $\mathbb{Z}_p$.
Even worse, the Mahler expansions need not even converge because if $x$ is not in $\mathbb{Z}_p$, the binomial coefficient values may have negative valuation.
EDIT: As Kevin Buzzard and dke suggest, one can give a positive answer if your question is interpreted differently. The point of this edit is to make a few explicit remarks in these two directions.
1) If it is known in advance that $f \colon \mathcal{O} \to \mathbb{C}_p$ is represented by a single convergent power series, then the Mahler expansion of $f|_{\mathbb{Z}_p}$ converges to $f$ on all of $\mathcal{O}$. This can be deduced from the theorem that a continuous function $\mathbb{Z}_p \to \mathbb{C}_p$ is analytic if and only if the Mahler expansion coefficients $a_n$ satisfy $a_n/n! \to 0$ (see Theorem 54.4 in Ultrametric calculus: an introduction to $p$-adic analysis by W. H. Schikhof).
2) If one chooses a $\mathbb{Z}_p$-basis of $\mathcal{O}$, then $f$ can be interpreted as a continuous function $\mathbb{Z}_p^r \to \mathbb{C}_p$, and any such function has a multivariable Mahler expansion
$$\sum a_n \binom{x_1}{n_1} \cdots \binom{x_r}{n_r},$$
where the sum is over tuples $n=(n_1,\ldots,n_r)$ with $n_i \in \mathbb{Z}_{\ge 0}$, and $a_n \to 0$ $p$-adically.
Best Answer
First a short answer. I don't think one can say that the commutative analytic side is known, as you do. It is fully known only in the cyclotomic $\mathbb{Z}_{p}$ situation, assuming the ETNC and in the crystalline case (see below). The answer to your "why is it so hard" question is: because $p$-adic $L$-functions are linked with $B_dR$ and so require subtle knowledge of $p$-adic Hodge theory which is lacking in the non-commutative case (and also mostly in the general commutative case). The answer to your technical question is: because in the ordinary case, there is a concrete incarnation of $D_{dR}$ which allows for a definition of the required trivialization. Outside the ordinary case, no such concrete incarnation is known and so things are two orders of magnitude harder, already in the commutative case (again, see below for some more details).
In both the commutative and non-commutative situation, if you want to construct a $p$-adic $L$-function in the style of Kato, Perrin-Riou, Colmez (basically in the cyclotomic $\mathbb{Z}_{p}$-extension case) and Fukaya-Kato, CFKSV (in the non-commutative but still number field tower case) you need two things: one is an equivariant basis of the fundamental line (more or less the determinant of the motivic cohomology of your motive, or more concretely some sort of Euler system), the other is what is usually called a reciprocity law or Coleman map or $\epsilon$-isomorphism to transform this into a "function" or "measure" or "element in a localized $K_{1}$" (depending what you mean by $p$-adic $L$-function).
Slightly more precisely and in a more technical language, in order to formulate a conjectural setting allowing for the existence of a $p$-adic $L$-function, you need a trivialization of some complexes of Galois cohomology which is compatible with "evaluation at characters" (without this compatibility, there can be no interpolation property worth its name). This trivialization involves very subtle properties of the ring $B_{dR}$.
Now, in the cyclotomic $\mathbb{Z}_{p}$-extension case, such a trivialization is known to exist for any crystalline motive thanks to very deep results of many people, but most notably Perrin-Riou, Kato-Kurihara-Tsuji, Colmez and Benois-Berger. So in that case, if you $assume$ the Equivariant Tamagawa Number Conjecture, then the $p$-adic $L$-function is well-defined. This is only in this (very limited) sense that the analytic object you refer to in your question is well-defined.
In the general commutative case, very little is known, because one has to consider the $D_{dR}$ of Galois representations with coefficients in rings of large dimension and this is extremely hard though spectacular progresses are made each day in that respect. In the good ordinary tower of number fields case, be it commutative or non-commutative, the required trivialization is known because in that case there is a "concrete incarnation" of $D_{dR}$.
Finally, in the general non-commutative case, almost nothing is known because, as far as I know, the collective knowledge we have on the behaviour of $D_{dR}$ for non-commutative rings of large dimension is very far from what would be required only to formulate a question.