Full Stage Lay Summary We will test a new approach to a form of heart failure (HF) with no current treatment proven to reduce death rates or hospitalisations. Over a third of HF cases have preserved ejection fraction (HFPEF) often on a background of high blood pressure (BP). These "stiff" hearts pump strongly but fill inefficiently resulting in poor exercise capacity and high death rates. Treatments that help when heart pumping action is poor are of no benefit in HFPEF. Recently a simple catheter procedure removing excess nerve signals to and from the kidneys ("renal denervation"; RDN) has been able to reduce BP in patients with high BP resistant to multi-drug treatment. Through removing excess nervous drive to the kidneys, heart and circulation this treatment has promise in HF. We will compare effects of RDN and standard medical treatment on heart function, exercise capacity and quality of life in 144 patients with HFPEF.