I thought it would be very easy to prove, but in fact, I did not manage to prove or disprove this fact: the sequence of polynomials $$\left(\prod_{j=0}^k\big(1-2^{2^j}X\big)\right)_{k\in\mathbb N}$$ converges in $\mathbb Q_2[[X]]$. When considered as function in $\mathbb Q_2$ it is very easy to see that the sequence converges towards a function defined over $\mathbb Q_2$, but I do not any idea when the sequence is considered as formal object.