Example of a differentiable $f$ and $(x^*,y^*)$ where $x^*=\arg\min_x f(x,y^*)$, $y^*=\arg\min_y f(x^*,y)$, yet $(x^*,y^*)\neq \arg\min_{x,y}f(x,y)$?

Question

More generally, could there be a point $x^*=(x_1^*,...,x_n^*)$ that minimizes a differentiable function $f: \mathbb{R^n} \to \mathbb{R}$ along each coordinate, i.e., $x_i^*=\arg\min_{x_i} f(x_1^*,...,x_{i-1}^*, x_i, x_{i+1}^*,...,x_n^*)$ (note this is a stronger condition than $x^*$ being stationary) , yet $x^*$ isn't the global minimum?

I suspect the answer is yes, but I'm having trouble coming up with a proof or example.