The question was a bit long for the title, so let me explain what I mean here. Let $k$ be some field (ideally, a number field). Let $X$ be a curve that is defined over $k$, such that it has a $k$-point. Let $K$ be an infinitely generated algebraic extension of $k$. Does this imply that $X$ has infinitely many $K$-points?


###Motivation
The canonical example I have of a field which is infinitely generated over its prime field, together with a curve, such that the curve doesn't have infinitely many rational points is: $x^2+y^2=-1$ where the field is $\mathbb{R}$. However in this case, it is not an infinitely generated algebraic extension of a field for which this curve does have a rational point.