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It might be noteworthy that the ring $\mathbb{Z}_p[[T]]$ is the Iwasawa algebra of the additive group $\mathbb{Z}_p$, and there is a nice structure theory for finitely generated modules over it. See the linked Wikipedia article, and the references therein, as well as e.g. chapters 7 and 13 of Washington's Introduction to Cyclotomic Fields or chapter 5 of Lang's Cyclotomic Fields (and for a much more general account, Bourbaki's Commutative Algebra ch. VII §4). However, this structure theory works "up to pseudo-isomorphism" i.e. up to so-called "pseudo-null" modules which in this case are exactly the ones with finite cardinality. All subquotients which are finite-dimensional $\mathbb{F}_p$-vector spaces are in this class; on the other hand, the rank is invariant under pseudo-isomorphism. So it is highly unlikely that the dimensions in your question will give you any non-trivial information about the rank of your module, as illustrated by abx's answer.