  Let $v$ be a finite place of $\mathbb{F}_q(\theta)$. We show that algebraic
relations of $v$-adic arithmetic gamma values over $\mathbb{F}_q(\theta)$ are
explained by the standard functional equations together with Thakur's analogue
of the Gross-Koblitz formula. A key step is working out a formula expressing
$v$-adic crystalline-de Rham periods of Carlitz motives with Complex
Multiplication as products of these special gamma values (a $v$-adic
Chowla-Selberg formula). The algebraic independence of those periods in
question results from determining the dimension of the motivic Galois groups,
adapting and further developing existing tools.
