  In this paper, we study $\lambda \phi^4$ scalar field theory defined on the
unramified extension of p-adic numbers ${\mathbb Q}_{p^n}$. For different
``space-time'' dimensions $n$, we compute one-loop quantum corrections to the
effective potential. Surprisingly, despite the unusual properties of
non-Archimedean geometry, the Coleman-Weinberg potential of p-adic field theory
has a structure very similar to that of its real cousin. We also study two
formal limits of the effective potential, $p \rightarrow 1$ and $p \rightarrow
\infty$. We show that the $p\rightarrow 1$ limit allows to reconstruct the
canonical result for real field theory from the p-adic effective potential and
provide an explanation of this fact. On the other hand, in the
$p\rightarrow\infty$ limit, the theory exhibits very peculiar behavior with
emerging logarithmic terms in the effective potential, which has no analog in
real theories.
