Let
p
p
be a prime, and let
K
K
be a finite extension of
Q
p
\mathbf {Q}_p
, with absolute Galois group
G
K
\mathcal {G}_K
. Let
π
\pi
be a uniformizer of
K
K
and let
K
∞
K_\infty
be the Kummer extension obtained by adjoining to
K
K
a system of compatible
p
n
p^n
-th roots of
π
\pi
, for all
n
n
, and let
L
L
be the Galois closure of
K
∞
K_\infty
. Using these extensions, Caruso has constructed é tale
(
ϕ
,
τ
)
(\phi ,\tau )
-modules, which classify
p
p
-adic Galois representations of
K
K
. In this paper, we use locally analytic vectors and theories of families of
ϕ
\phi
-modules over Robba rings to prove the overconvergence of
(
ϕ
,
τ
)
(\phi ,\tau )
-modules in families. As examples, we also compute some explicit families of
(
ϕ
,
τ
)
(\phi ,\tau )
-modules in some simple cases.