Let
E
E
be a non-Archimedean Banach space over a non-Archimedean locally compact non-trivially valued field
\mathbb{K}:=(\mathbb{K},|.|)
K
:
=
(
K
,
∣
.
∣
)
. Let
E''
E
′
′
be its bidual and
M
M
a bounded set in
E
E
. We say that
M
M
is
\varepsilon
ε
-weakly relatively compact if
\ \overline{M}^{\sigma (E'',E')}\subset E+B_{E^{\prime \prime },\varepsilon}
M
‾
σ
(
E
′
′
,
E
′
)
⊂
E
+
B
E
′
′
,
ε
, where
B_{E^{\prime \prime },\varepsilon }
B
E
′
′
,
ε
is the closed ball in
E''
E
′
′
with the radius
\varepsilon \geq 0
ε
≥
0
. In this paper we describe measures of noncompactness
\gamma,
γ
,
k
k
and De Blasi measure
\omega
ω
. We show that
\gamma \left( M\right) \leq k\left( M\right) \leq \omega \left( M\right) =\omega (acoM)\leq \frac{1}{\left\vert \rho \right\vert }\gamma \left( M\right),
γ
(
M
)
≤
k
(
M
)
≤
ω
(
M
)
=
ω
(
a
c
o
M
)
≤
1
∣
ρ
∣
γ
(
M
)
,
where
\rho
ρ
(
\left\vert \rho \right\vert <1)
∣
ρ
∣
<
1
)
is an uniformizing element in
\mathbb{K}
K
, and
\omega (M)=\sup \{\overline{\lim_{m}}\,\,\,dist\left( x_{m},\left[ x_{1},\dots,x_{m-1}\right] \right):\left( x_{m}\right) \subset M
ω
(
M
)
=
sup
{
lim
m
‾
d
i
s
t
(
x
m
,
[
x
1
,
…
,
x
m
−
1
]
)
:
(
x
m
)
⊂
M
\}
}
; the latter equality is purely non-Archimedean. In particular, assuming
\left\vert \mathbb{K}\right\vert =\{||x||:x\in E\},
∣
K
∣
=
{
∣
∣
x
∣
∣
:
x
∈
E
}
,
we prove that the absolutely convex hull
acoM
a
c
o
M
of a
\varepsilon -
ε
−
weakly relatively compact subset
M
M
in
E
E
is
\varepsilon -
ε
−
weakly relatively compact. In fact we show that in this case for a bounded set
M
M
in
E
E
we have
\gamma \left( M\right) =\gamma \left( acoM\right) =k\left( M\right) =k(acoM)=\omega \left( M\right)
γ
(
M
)
=
γ
(
a
c
o
M
)
=
k
(
M
)
=
k
(
a
c
o
M
)
=
ω
(
M
)
, Note that the above equalities fail in general for real Banach spaces by results of A. S. Granero [An extension of the Krein-Smulian theorem, Rev. Mat. Iberoam. 22 (2006) 93–100] and K. Astala and H. O. Tylli [Seminorms related to weak compactness and to Tauberian operators, Math. Proc. Cambridge Philos. Soc. 107 (1990) 367–375]. Most proofs are strictly non-Archimedean. A non-Archimedean variant of another quantitative Krein's theorem due to Fabian, Hajek, Montesinos and Zizler is also provided, see Corollary 9.