Suppose $F(x,y) \in \mathbb{Z}_p[[x,y]]$ is a formal group over $\mathbb{Z}_p$. I denote with $G(-)$ the corresponding functor, so that $G(K)$ will denote for me the maximal ideal of $O_K$ equipped with the operation given by $F$, where $K$ is a finite extension of $\mathbb{Q}_p$.
Can $\mathbb{F}_p \otimes_{\mathbb{Z_p}}\text{Tor}(G(K))$ (by Tor(-) i mean the torsion) become arbitrarily large for suitable finite extensions of $\mathbb{Q}_p$? If the answer is no, that is eventually constant (that is, constant on all extensions of a suitable finite extension), can this constant be any nonnegative integer?
The baby example I have in mind is the multiplicative group with rank 1.