Packer system with a spring and ratchet mechanism for wellbore operations

A packer system includes a retainer positionable adjacent to a packing element and coupleable to a cylinder of a downhole tool in a wellbore. The retainer is moveable in a first direction along a mandrel of the downhole tool to compress the packing element to a compressed state and generate a seal in the wellbore or to set a slip in response to an applied pressure. The packer system includes a piston moveable in a second direction along the mandrel and configured to compress a spring in response to the applied pressure. The packer system includes an interfacing element configured to couple the cylinder to the piston subsequent to the cylinder moving a predefined amount. The packer system includes a ratchet mechanism configured to prevent movement of the cylinder in the second direction.