$\hat\beta$ is unique exactly when $X$ has full column rank

Consider the multiple regression set-up $\{Y,X\beta,\sigma^2I\}$ with $p$ regressors. The ordinary least squares estimator (OLSE) $\hat\beta$ is the vector which minimizes $(Y-X\beta)'(Y-X\beta)$. Which of the following statements is not true under this set-up?