Reduction of the Manin map modulo $p$

For an abelian variety $A$ over a function field $K$ of characteristic zero, Manin defined a remarkable additive map $A(K) \ra V$, where $V$ is a vector space over $K$. We define an analogue of this map in the case of function fields of characteristic $p$. We then prove that the reduction modulo $p$ of the Manin map in characteristic zero is the derivative of the Manin map in characteristic $p$ and that the kernel of the Manin map in characteristic $p$ is the group of points divisible by $p$.