I have sent Squarespace a request and linked to this thread. Let's see what they say...
In the meantime, there is a workaround. It is far from perfect, but it might help some people.
1) Insert a "Code block". As a first step you can leave it with just the default "Hello world".
2) Inspect the existing element (tool available at least in Chrome & Firefox when right-clicking on an element).
3) In the inspector, select the complete block you want to duplicate.
4) Right-click > copy Outer HTML
5) Paste in the Code block you inserted in step 1.
Any further modifications to the new blocks will have to be done by coding though...