diff --git a/sphinx/builders/linkcheck.py b/sphinx/builders/linkcheck.py
index f4299d14c..479a5b877 100644
--- a/sphinx/builders/linkcheck.py
+++ b/sphinx/builders/linkcheck.py
@@ -18,7 +18,7 @@ from urllib.parse import unquote, urlparse, urlsplit, urlunparse
 
 from docutils import nodes
 from requests import Response
-from requests.exceptions import ConnectionError, HTTPError, TooManyRedirects
+from requests.exceptions import ConnectionError, HTTPError, SSLError, TooManyRedirects
 
 from sphinx.application import Sphinx
 from sphinx.builders.dummy import DummyBuilder
@@ -333,6 +333,10 @@ class HyperlinkAvailabilityCheckWorker(Thread):
                     del response
                     break
 
+                except SSLError as err:
+                    # SSL failure; report that the link is broken.
+                    return 'broken', str(err), 0
+
                 except (ConnectionError, TooManyRedirects) as err:
                     # Servers drop the connection on HEAD requests, causing
                     # ConnectionError.
@@ -361,7 +365,7 @@ class HyperlinkAvailabilityCheckWorker(Thread):
                     continue
 
                 except Exception as err:
-                    # Unhandled exception (intermittent or permanent); report that the
+                    # Unhandled exception (intermittent or permanent); report that
                     # the link is broken.
                     return 'broken', str(err), 0
 
diff --git a/tests/test_build_linkcheck.py b/tests/test_build_linkcheck.py
index f4229cccd..d7c925e7c 100644
--- a/tests/test_build_linkcheck.py
+++ b/tests/test_build_linkcheck.py
@@ -18,6 +18,7 @@ import pytest
 
 from sphinx.builders.linkcheck import HyperlinkAvailabilityCheckWorker, RateLimit
 from sphinx.testing.util import strip_escseq
+from sphinx.util import requests
 from sphinx.util.console import strip_colors
 
 from .utils import CERT_FILE, http_server, https_server
@@ -394,7 +395,9 @@ class OKHandler(http.server.BaseHTTPRequestHandler):
 def test_invalid_ssl(app):
     # Link indicates SSL should be used (https) but the server does not handle it.
     with http_server(OKHandler):
-        app.build()
+        with mock.patch("sphinx.builders.linkcheck.requests.get", wraps=requests.get) as get_request:
+            app.build()
+            assert not get_request.called
 
     with open(app.outdir / 'output.json', encoding='utf-8') as fp:
         content = json.load(fp)