diff --git a/panel/chat/interface.py b/panel/chat/interface.py
index 3ef6e68e77..d23a154f26 100644
--- a/panel/chat/interface.py
+++ b/panel/chat/interface.py
@@ -292,8 +292,8 @@ def _init_widgets(self):
                     sizing_mode="stretch_width",
                     max_width=show_expr.rx.where(90, 45),
                     max_height=50,
-                    margin=(5, 5, 5, 0),
-                    align="start",
+                    margin=(0, 5, 0, 0),
+                    align="center",
                     visible=visible
                 )
                 if action != "stop":
diff --git a/panel/chat/message.py b/panel/chat/message.py
index 9996c5a5b9..229c6f3a22 100644
--- a/panel/chat/message.py
+++ b/panel/chat/message.py
@@ -277,6 +277,7 @@ def _build_layout(self):
                 self.chat_copy_icon,
                 stylesheets=self._stylesheets,
                 sizing_mode="stretch_width",
+                css_classes=["header"]
             ),
             self._center_row,
             self._timestamp_html,
diff --git a/panel/dist/css/chat_message.css b/panel/dist/css/chat_message.css
index 692dcc4cfa..c85c9753ba 100644
--- a/panel/dist/css/chat_message.css
+++ b/panel/dist/css/chat_message.css
@@ -47,6 +47,10 @@
   max-width: calc(100% - 80px);
 }
 
+.header {
+  width: fit-content;
+}
+
 .name {
   font-size: 1em;
   margin-bottom: 0px;
@@ -88,7 +92,7 @@
 .timestamp {
   color: #a9a9a9;
   display: flex;
-  margin-top: 0px;
+  margin-top: 3px;
 }
 
 .markdown {