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, 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, Pure and Applied Math Quarterly 5 (2009), number 1, 375-434