I haven't listened to radio 4 for a while (news got too depressing and had to stop listening), but although there are a lot more examples of this on BBC1 television channels, Radio 4 is far from immune. I listened when they had a 'climate debate' between a number of climate scientists talking about man made global warming and the well known climate scientist Nigel Lawson, who was saying there was no evidence of it. Yes, that Nigel Lawson (ex thatcher CoE and Prominant brexiter who then failed to get his french citizenship after we left). Both sides were given equal weigh so they were 'Balanced'.
I hope if you say you have never heard anything like that, you have only just started listening and it has got better, but it was certainly not uncommon back a few years ago.
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Oh yes, here is a bit on that: https://www.theguardian.com/environment/2018/apr/09/bbc-radio-4-broke-impartiality-rules-in-nigel-lawson-climate-change-interview
and I note that they had been done for that previously, but there are other examples of this sort of behaviour. As I said, radio 4 is about the best. BBC1 news is about the worst,