Possibly. I don't know enough about the subject to comment further, but perhaps
@The Ham does?
There's actually very little which would stop the highway authority from narrowing most roads if they felt minded to do so.
Without knowing the road in question it would be hard to speculate, however subject to the mix of traffic it is possible to have roads of 7.3m in width for areas with high HGV content especially when they are turning (such as industrial estates) 6.1m in width (more limited HGV use but used by buses), where there's no buses or can go down to 5.5m (as above but no bus route), 4.8m is also acceptable where HGV use would be infrequent (typically low enough that it's unlikely that 2 HGV's would meet) whilst 4.1m allows two cars to pass but would need junctions and widenings to allow cars to pass the fairly rare HGV.
The width of the road doesn't normally officially come into play when determining speed limits, however you certainly wouldn't want a 4.1m wide road on a 60mph limit (unless other factors were limiting vehicle speeds). The reason for this is that few roads actually meet the official design standards unless they are new build roads (i.e. within the last 60 years).
By the sounds of things the reply is that of a council which doesn't want to do anything, if you really wanted to call their bluff you could ask them for the copy of the Road Safety Audit which confirms that the provision of a footway would cause risk to traffic (it's almost certainly that no such document exists as no one has even looked at what the provision of a footway would even look like).
It's also worth noting that (assuming that there's likely to be a fairly strong demand for a walking route, such as to a station, school, etc.) that semi informal paths (i.e. with stones to stop then getting muddy but the land is leased from the farmer rather than adopted by the council) can be implemented.
There's an ongoing cost in the lease, however the annual cost of leasing a 3m strip of farm land isn't all that high as you're basically paying for the loss of crop growing capacity. It's almost certainly something that a Parish Council could reasonably afford if the County Council would fund the upfront costs for the works to the path and the required extra fencing.
As it's got a lease on it, it doesn't become a right of way so there's nothing stopping the landowner selling the land on, and although any development would almost certainly need to prove a replacement route; such a route could be provided anywhere suitable rather than along that exact route without the need to consult on diverting a public right of way.
Farmers, of course, may still not want to do so, however it does ensure a guaranteed income for very little effort for a strip of land that's unlikely to be overly productive anyway. However worth the likely extra bit of wildlife refuge it's possible that it could encourage the sorts of things which are the predictors of things which harm their crops/livestock.