  Let $p$ be a prime number. There are properties called ``overconvergence''
and ``$F$-analyticity'' for $p$-adic Galois representations of a $p$-adic field
$F$. By Berger's work, it is known that $F$-analyticity is stricter than
overconvergence. In this article, we show that, in many cases, an
overconvergent Galois representation is $F$-analytic up to a twist by a
character. This result emphasizes the necessity of the theory of
$(\varphi,\Gamma)$-modules over the multivariable Robba ring, by which we
expect to study all $p$-adic Galois representations.
