There is no legal duty on anyone in England to care for another person. So the answer is no, not if a carer can show they have CLEARLY told the local authority or relevant funder that they are not willing or not able to do so. Ignoring the carer’s stance or withdrawal would be a failure to meet the needs, in breach of duty. Moreover, even when a person is willing, it may be that objectively speaking, they cannot be regarded as able, and the authority must make a rational decision about that and step up in that scenario.
In the real world, a carer’s silence may, in practice, be interpreted as willingness to continue.
This CASCAIDr link is a summary of a case in which a local authority asserted that if a carer was providing care, then they must be willing to do so.
The judge stated that while the concept of willingness was in the hands of the carer, a professional’s objective view of a person’s ability to continue caring was a public law decision and as such, where a local authority or funder ignored or ‘hid from’ clear evidence that a carer could not reasonably be expected to carry on providing care, due to the impact upon them or their mental or physical health – then it could be found that that not made a rational decision and had ignored relevant considerations.