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} .