Her daughters, actresses Emily and Daisy Head, announced her death, describing it as "immensely shocking" and praising her as "extraordinary, kind and talented".
Ms Fisher ran Tilley Farm, a 90-acre property in Somerset, and was a dedicated advocate for animal welfare.
She served as an ambassador for Battersea Dogs And Cats Home and a patron of Holly Hedge Animal Sanctuary.
Tributes from Battersea CEO Peter Laurie and Holly Hedge founder Pauline Leeson highlighted her compassionate nature and significant contributions to helping animals.
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