diff --git a/.github/workflows/ci.yaml b/.github/workflows/ci.yaml
index daafc8c..395d2e5 100644
--- a/.github/workflows/ci.yaml
+++ b/.github/workflows/ci.yaml
@@ -8,15 +8,19 @@ jobs:
             matrix:
                 include:
                     - arch: amd64
+                      manylinux_arch: x86_64
                       docker_image: debian:buster
 
                     - arch: arm64
+                      manylinux_arch: aarch64
                       docker_image: arm64v8/debian:buster
 
                     - arch: ppc64le
+                      manylinux_arch: ppc64le
                       docker_image: ppc64le/debian:buster
 
                     - arch: s390x
+                      manylinux_arch: s390x
                       docker_image: s390x/debian:buster
 
         env:
@@ -46,6 +50,9 @@ jobs:
               # sure it installs in a clean image without any hidden dependencies.
               run: docker run --rm -v $(pwd):/mnt:rw ${{ matrix.docker_image }} /mnt/ci/docker-deb-test
 
+            - name: Build wheels
+              run: sudo make python-dists-${{ matrix.manylinux_arch }}
+
             - name: Upload build artifacts
               uses: actions/upload-artifact@v2
               with:
diff --git a/Makefile b/Makefile
index f98c97d..1ddbb37 100644
--- a/Makefile
+++ b/Makefile
@@ -30,14 +30,17 @@ release: python-dists
 		> sha256sums
 
 .PHONY: python-dists
-python-dists: VERSION.h
+python-dists: python-dists-x86_64 python-dists-aarch64 python-dists-ppc64le python-dists-s390x
+
+.PHONY: python-dists-%
+python-dists-%: VERSION.h
 	python setup.py sdist
 	docker run \
 		--user $$(id -u):$$(id -g) \
-		-v $(PWD)/dist:/dist:rw \
-		quay.io/pypa/manylinux1_x86_64:latest \
+		-v `pwd`/dist:/dist:rw \
+		quay.io/pypa/manylinux2014_$*:latest \
 		bash -exc ' \
-			/opt/python/cp35-cp35m/bin/pip wheel --wheel-dir /tmp /dist/*.tar.gz && \
+			/opt/python/cp38-cp38/bin/pip wheel --wheel-dir /tmp /dist/*.tar.gz && \
 			auditwheel repair --wheel-dir /dist /tmp/*.whl --wheel-dir /dist \
 		'
 
diff --git a/requirements-dev.txt b/requirements-dev.txt
index 2dac807..49f54fd 100644
--- a/requirements-dev.txt
+++ b/requirements-dev.txt
@@ -1,3 +1,6 @@
 pre-commit>=0.5.0
 pytest
-pytest-timeout
+# TODO: This pin is to work around an issue where the system pytest is too old.
+# We should fix this by not depending on the system pytest/python packages at
+# some point.
+pytest-timeout<2.0.0