Prosthetic heart valve devices and associated methods for percutaneous or transcatheter heart valve replacement are disclosed herein. A heart valve prosthesis configured in accordance herewith includes a frame having a valve support and a plurality of support arms extending therefrom. The plurality of support arms may include a main support arm configured to extend from the valve support for capturing at least a portion of a valve leaflet of a native heart valve therebetween when the valve prosthesis is in an expanded configuration and deployed within the native heart valve. In addition, the plurality of support arms may include multiple supplemental support arms disposed about the circumference of the valve support that when deployed in the expanded configuration are configured to at least partially engage subannular tissue at the native heart valve.