  In the present paper we provide a new construction of measure, called
$p$-adic quasi Gibbs measure, for countable state of $p$-adic Potts model on
the Cayley tree. Such a construction depends on a parameter $\frak{p}$ and
wights. In particular case, i.e. if $\frak{p}=\exp_p$, the defined measure
coincides with $p$-adic Gibbs measure. In this paper, under some condition on
weights we establish the existence of $p$-adic quasi Gibbs measures associated
with the model. Note that this condition does not depend on values of the prime
$p$. An analogues fact is not valid when the number of spins is finite.
