  We consider an infinite extension $K$ of a local field of zero characteristic
which is a union of an increasing sequence of finite extensions. $K$ is
equipped with an inductive limit topology; its conjugate $\bar{K}$ is a
completion of $K$ with respect to a topology given by certain explicitly
written seminorms. The semigroup of measures, which defines a stable-like
process $X(t)$ on $\bar{K}$, is concentrated on a compact subgroup $S\subset
\bar{K}$. We study properties of the process $X_S(t)$, a part of $X(t)$ in $S$.
It is shown that the Hausdorff and packing dimensions of the image of an
interval equal 0 almost surely. In the case of tamely ramified extensions a
correct Hausdorff measure for this set is found.
