  Neretin constructed an analogue of the Hua measures on the infinite $p$-adic
matrices $Mat\left(\mathbb{N},\mathbb{Q}_p\right)$. Bufetov and Qiu classified
the ergodic measures on $Mat\left(\mathbb{N},\mathbb{Q}_p\right)$ that are
invariant under the natural action of $GL(\infty,\mathbb{Z}_p)\times
GL(\infty,\mathbb{Z}_p)$. In this paper we solve the problem of ergodic
decomposition for the $p$-adic Hua measures introduced by Neretin. We prove
that the probability measure governing the ergodic decomposition has an
explicit expression which identifies it with a Hall-Littlewood measure on
partitions. Our arguments involve certain Markov chains.
