Q:

Is there an analogue of Mordell-Weil theorem for other fields?

The Mordell-Weil theorem states that for an abelian variety $A$ over a number field $K$ the group of $K$-rational points of $A$ is finitely generated and abelian.
What if $K$ is not a number field, e. g. Q$_p$ or a transcendental extension of $\mathbb Q$? 

A:

For $K=\Bbb Q_p$ there will be uncountably many $K$-rational points.
The exact structure of $K$-rational points will depend on the
reduction type of the curve, but points close to the base-point $O$ are
in essence parameterised by the formal group of the curve. See Silverman's book for much more detail.

