Example of convex function with global min that is Lipschitz but does not have Lipschitz gradients

Question

In gradient descent, when optimizing a convex function with a global minimum, one often assumes either that

1. the function is Lipschitz, or

2. that its gradients are Lipschitz.

There are examples where 2 does not imply 1. Are there any examples where 1 does not imply 2 (restricted to convex case with a global min)?