  In the present paper we consider countable state $p$-adic Potts model on the
Cayley tree. A construction of $p$-adic Gibbs measures which depends on weights
$\l$ is given, and an investigation of such measures is reduced to examination
of an infinite-dimensional recursion equation. Studying of the derived equation
under some condition on weights, we prove absence of the phase transition. Note
that the condition does not depend on values of the prime $p$, and an analogues
fact is not true when the number of spins is finite. For homogeneous model it
is shown that the recursive equation has only one solution under that condition
on weights. This means that there is only one $p$-adic Gibbs measure $\m_\l$.
The boundedness of the measure is also established. Moreover, continuous
dependence the measure $\m_\l$ on $\l$ is proved. At the end we formulate one
limit theorem for $\m_\l$.
