Most universities will have criteria for deciding what happens once you reach four years of continuous employment. They don't have to give you a proper permanent contract.
Most commonly for research staff working exclusively on externally funded research projects, the University can use an objective criteria for exclusion, one of which is that you are working on a project with time limited funding and there is not sufficient evidence to show that there will be any funding available once the funding for that specific project finishes. You can push back and say that your successive fixed-term contracts demonstrate that there is ongoing funding around and a reasonable chance that you will win some of it, so continuing to fund your post. This is easier to claim if you are winning funding for your own research as at least a named researcher on proposals or if you are working on projects with a very prolific and established PI who can provide evidence of their ability to continually win funding and hence support your post.
If the university decide not to give you a permanent contract, they can give you an open-ended contract where you have certain rights around redundancy but basically once your funding comes to an end, you are made redundant unless you have managed to source further funding. In certain circumstances, they can refuse the request entirely and keep you on a fixed-term contract.
I do know researchers on proper permanent contracts, but as far as I know, this has come about in one of two ways. One is where they are members of a research institute / centre/ group and the centre collectively maintains an income on a sustained basis sufficient to support the salaries, etc., of all their staff. The other is where someone had a permanent teaching and research contract and moved onto a permanent research only contract because they were consistently winning enough research funding to demonstrate that they can, on an ongoing basis, fund their post on research income alone.
In terms of doing other stuff and progressing, this is very difficult when your post is entirely funded by an externally funded research grant because you are funded to fulfil the terms of that grant. Ideally, the grant you are working on should include time for you to do things like publishing and professional development, but this is far from always being the case. The people in research centres with permanent posts are able to have time for other activities because their collective funding comes with a high profit margin that they can reinvest in the development of their staff alongside income the centre as a whole gets from other income sources, including the REF. This is not really the case for lone researchers outside these centres, unless their research has some kind of strategic or reputation benefit to the University. The bottom line is that everything has to be paid for and you have to be able to show where the money is coming from.