Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism
$$DR(V) \rightarrow H^1_e(K, V)$$
and state that (Bloch-Kato, Example 3.10.1) for a commutative formal lie group $G$ of finite height over $\mathcal{O}_K$, their exponential map $$T_p \otimes \mathbb{Q} \xrightarrow{exp_{BK}} DR(T_p \otimes \mathbb{Q})$$(for $T_p$ the $p$-adic tate module of G) agrees with the "exponential map in the classical sense"
$$\textrm{tan}(G_K) \xrightarrow{exp_{classical}} G(O_K) \otimes \mathbb {Q} $$
That is, a certain diagram commutes (Bloch-Kato, Example 3.10.1).
$\textbf{Question:}$ Can you give me a reference defining the exponential map for the formal lie group associated to an abelian variety $A$ over $K$?
If necessary, feel free to assume $A$ has semistable reduction over $K$. Is this contained in the second half of Serre's "Lie groups and Lie algebras", or Fontaine's "Groupes p-divisibles sur les corps locaux"?