We show that the non-archimedean version of Grothendieck's theorem about weakly compact sets for
C(X,\mathbb{K})
C
(
X
,
K
)
, the space of continuous maps on
X
X
with values in a locally compact non-trivially valued non-archimedean field
\mathbb{K}
K
, fails in general. Indeed, we prove that if
X
X
is an infinite zero-dimensional compact space, then there exists a relatively compact set
H:=\{g_{n}:n\in \mathbb{N}\}\subset C(X,\mathbb{K})
H
:
=
{
g
n
:
n
∈
N
}
⊂
C
(
X
,
K
)
in the pointwise topology
\tau _{p}
τ
p
of
C(X,\mathbb{K})
C
(
X
,
K
)
which is not
w-
w
−
relatively compact, i.e. compact in the weak topology of
C(X,\mathbb{K})
C
(
X
,
K
)
, such that all
\Vert g_{n}\Vert =1
∥
g
n
∥
=
1
and
\gamma (H):=\sup \{|\lim_{m}\lim_{n}f_{m}(x_{n})-\lim_{n}\lim_{m}f_{m}(x_{n})|:(f_{m})_{m} \subset B,(x_{n})_{n}\subset H\}>0
γ
(
H
)
:
=
sup
{
∣
lim
m
lim
n
f
m
(
x
n
)
−
lim
n
lim
m
f
m
(
x
n
)
∣
:
(
f
m
)
m
⊂
B
,
(
x
n
)
n
⊂
H
}
>
0
, where
B
B
is the closed unit ball in the dual
C(X,\mathbb{K})^{\ast }
C
(
X
,
K
)
∗
and the involved limits exist. The latter condition
\gamma (H)>0
γ
(
H
)
>
0
shows in fact that a quantitative version of Grothendieck's theorem for real spaces (due to Angosto and Cascales) fails in the non-archimedean setting. The classical Krein and Grothendieck's theorems ensure that for any compact space
X
X
every uniformly bounded set
H
H
in a real (or complex) space
C(X)
C
(
X
)
is
\tau _{p}
τ
p
-relatively compact if and only if the absolutely convex hull
\operatorname{aco}H
aco
H
of
H
H
is
\tau _{p}
τ
p
-relatively compact. In contrast, we show that for an infinite zero-dimensional compact space
X
X
the absolutely convex hull
\operatorname{aco}H
aco
H
of a
\tau _{p}-
τ
p
−
relatively compact and uniformly bounded set
H
H
in
C(X,\mathbb{K})
C
(
X
,
K
)
needs not be
\tau _{p}-
τ
p
−
relatively compact for a locally compact non-archimedean
\mathbb{K}
K
. Nevertheless, our main result states that if
H\subset C(X,\mathbb{K})
H
⊂
C
(
X
,
K
)
is uniformly bounded, then
\operatorname{aco}H
aco
H
is
\tau _{p}-
τ
p
−
relatively compact if and only if
H
H
is
w
w
-relatively compact.