diff --git a/zebra-network/src/peer_set/initialize.rs b/zebra-network/src/peer_set/initialize.rs
index aa3819ed..68312f1a 100644
--- a/zebra-network/src/peer_set/initialize.rs
+++ b/zebra-network/src/peer_set/initialize.rs
@@ -415,6 +415,16 @@ where
                     .instrument(handshaker_span),
                 );
             }
+
+            // Only spawn one inbound connection handshake per `MIN_PEER_CONNECTION_INTERVAL`.
+            // But clear out failed connections as fast as possible.
+            //
+            // If there is a flood of connections,
+            // this stops Zebra overloading the network with handshake data.
+            //
+            // Zebra can't control how many queued connections are waiting,
+            // but most OSes also limit the number of queued inbound connections on a listener port.
+            tokio::time::sleep(constants::MIN_PEER_CONNECTION_INTERVAL).await;
         }
     }
 }