Does $v=u^2$ force an analytic function to be constant?

Let \(f=u+iv\) be analytic on an open set \(U\subseteq\mathbb C\) and suppose \[ v=u^2. \] Can one conclude that \(f\) is constant on \(U\)?

If yes, prove it. If not, give a counterexample and modify the statement so that the corresponding constant-function conclusion is true, then prove the modified statement.