Let $K$ be an algebraic function field of characteristic $2$ with constant field $C_K$. Let $C$ be the algebraic closure of a finite field in $K$. Assume that $C$ has an extension of degree $2$. Assume that there are elements $u,x$ of $K$ with $u$ transcendental over $C_K$ and $x$ algebraic over $C(u)$ and such that $K=C_K(u,x)$. Then Hilbert's Tenth Problem over $K$ is undecidable. Together with Shlapentokh's result for odd characteristic this implies that Hilbert's Tenth Problem for any such field $K$ of finite characteristic is undecidable. In particular, Hilbert's Tenth Problem for any algebraic function field with finite constant field is undecidable.