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Let $\ell$ and $p$ be two distinct primes. We study the $p$-adic valuation ofthe number of spanning trees in an abelian $\ell$-tower of connectedmultigraphs. This is analogous to the classical theorem of Washington--Sinnotton the growth of the $p$-part of the class group in a cyclotomic$\mathbb{Z}_\ell$-extension of abelian extensions of $\mathbb{Q}$. Furthermore,we show that under certain hypotheses, the number of primes dividing the numberof spanning trees is unbounded in such a tower.

The non-$\ell$-part of the number of spanning trees in abelian $\ell$-towers of multigraphs