In this paper we continue our study of differentiation on a local field K. We define strong derivatives of fractional order
α
>
0
\alpha \, > \,0
for functions in
L
r
(
K
)
{L_r}(\textbf {K})
,
1
⩽
r
>
∞
1\, \leqslant \,r\, > \,\infty
. After establishing a number of basic properties for such derivatives we prove that the spaces of Bessel potentials on K are equal to the spaces of strongly
L
r
(
K
)
{L_r}(\textbf {K})
-differentiable functions of order
α
>
0
\alpha \, > \,0
when
1
⩽
r
⩽
2
1\, \leqslant \,r\, \leqslant \,2
. We then focus our attention on the relationship between these spaces and the generalized Lipschitz spaces over K. Among others, we prove an inclusion theorem similar to a wellknown result of Taibleson for such spaces over
R
n
{\textbf {R}^n}
.