A heart valve repair system includes a delivery sheath and an implant that includes a frame having a surface configured to contact an upstream surface of a native heart valve. First and second gripping members are coupled to the frame and each (1) includes first and second arms and (2) is configured to clamp a respective native leaflet. The implant is disposed in the sheath in a delivery state in which the frame defines a wall fully surrounding a central longitudinal axis of the implant. The distal end of the wall defines a distal opening of the frame. The distal end of the wall is disposed proximally to the entire first tissue-engaging surface of each of the gripping members and proximally to the entire second tissue-engaging surface of each of the gripping members. Other embodiments are also described.