Is it okay to work on a research project with a professor at a different university, in addition to main PhD project? In the context of a social sciences PhD in the UK, I want to know (i) whether it is permissible to work on  another research project; and (ii) whether there are any general practices, if I choose to work on a research project with a professor in a different university, in addition to my existing PhD research project with my main supervisor. The topics of my main PhD research project and the other project are closely linked.

Not sure if this is relevant, but both my main supervisor and the other professor at a different university frequently collaborate on projects.

Edit: I asked my advisor about this, and she stated it was fine (and we had to figure out details), but I could sense that she was very hesitant about it. I am not sure if I offended her by asking at all, but I am very keen to work on this other project.
In the UK the funding system is somewhat different to many other countries: PhD students are either self-funded, or receive stipends through Centers of Doctoral Training or from University funds. Since you are not funded from you professor's grant, you do not work for him/her. You work on your PhD project to become an independent researcher, and as such, you can (and sometimes expected to) make your own choice in how the project is executed.

Of course, a simple courtesy requires you to ask your supervisor first before collaborating with someone else. One obvious reason is that you probably don't fully know the relations between your professor and other colleagues and could potentially disclose some of your supervisor's research ideas to a colleague who is more of a competitor than a collaborator. Academia is a twisted world sometimes and it is although you have full right to manage your own PhD project, a bit of information and advice would never hurt.