A method of determining a presence of a leak in a wall of a production tubing string includes positioning an electrical submersible pump is positioned at the string's downhole end. Based on measurements from a pressure sensor positioned at a downhole location at a discharge end of the electrical submersible pump, it is determined that a discharge pressure has decreased over a specified time period. Based on measurements from a pressure sensor within a wellhead of the well, it is determined that a pressure within the production tubing string at the surface has decreased over the specified time period. Based on measurements from a flow-rate sensor at an uphole end of the production tubing string, it is determined the flow rate has decreased relative to a calculated expected flow rate. Based on a pressure integrity test initiated in response to these determinations, the presence of the leak is determined.