Short answer: yes.
Long answer: I can’t find anywhere in the Highway Code or elsewhere reputable that says that it is generally prohibited. That said, there will be specific signposted exceptions (bylaws may apply, perhaps).
However, when it comes to the England and Wales “public footpaths” mentioned earlier, there are reputable sources who will claim that it is not legal** to push a bike on a public footpath because a bike isn’t part of a pedestrian’s “usual accompaniment” (a web search for that phrase will get a whole range of views on that).
I suspect that there are people reading this who may actually be aware of any case law here; I certainly am not.
** Still not a lawyer, but a potential civil offence of trespass rather than a criminal one, I suspect.