Yes, there are $p$-adic analogues of (2). The case where $F$ is abelian over $\mathbf{Q}$ is known: see the paper

Manfred Kolster and Thong Nguyen Quang Do, *[Syntomic regulators and special values of p-adic L-functions][1]*, Invent. math. 133, 417-447 (1998).

From the abstract: "In this paper p-adic analogs of the Lichtenbaum Conjectures are proven for abelian number fields $F$ and odd prime numbers
$p$, which generalize Leopoldt's $p$-adic class number formula, and
express special values of $p$-adic $L$-functions in terms of orders of $K$-groups and higher $p$-adic regulators."

There's also been a lot of more recent work on this. For example, see Besser, De Jeu, Buckingham and Roblot, *[On the p-adic Beilinson conjecture for number fields][2]*, Pure and Applied Math Quarterly **5** (2009), number 1, 375-434


  [1]: http://www.researchgate.net/79e41511139fa3ac8a.pdf
  [2]: http://www.few.vu.nl/~jeu/abstracts/pBC.pdf