diff --git a/.github/workflows/check.yml b/.github/workflows/check.yml
index f2b1f237f..f7b8cbb9c 100644
--- a/.github/workflows/check.yml
+++ b/.github/workflows/check.yml
@@ -34,9 +34,11 @@ jobs:
           - 3.7
           - 3.6
           - 3.5
-          - pypy3
+          - pypy-3.6
+          - pypy-3.7-v7.3.6rc3
+          - pypy-3.8-v7.3.6rc3
           - 2.7
-          - pypy2
+          - pypy-2.7
         include:
           - { os: macos-latest, py: brew@py3 }
     steps:
diff --git a/docs/changelog/2182.bugfix.txt b/docs/changelog/2182.bugfix.txt
new file mode 100644
index 000000000..0f26a202b
--- /dev/null
+++ b/docs/changelog/2182.bugfix.txt
@@ -0,0 +1,2 @@
+Fixed path collision that could lead to a PermissionError or writing to system
+directories when using PyPy3.8 - by :user:`mgorny`.
diff --git a/src/virtualenv/create/via_global_ref/builtin/pypy/pypy3.py b/src/virtualenv/create/via_global_ref/builtin/pypy/pypy3.py
index 2e7e10c82..25d8e1680 100644
--- a/src/virtualenv/create/via_global_ref/builtin/pypy/pypy3.py
+++ b/src/virtualenv/create/via_global_ref/builtin/pypy/pypy3.py
@@ -28,7 +28,7 @@ class PyPy3Posix(PyPy3, PosixSupports):
     @property
     def stdlib(self):
         """PyPy3 respects sysconfig only for the host python, virtual envs is instead lib/pythonx.y/site-packages"""
-        return self.dest / "lib" / "python{}".format(self.interpreter.version_release_str) / "site-packages"
+        return self.dest / "lib" / "pypy{}".format(self.interpreter.version_release_str) / "site-packages"
 
     @classmethod
     def _shared_libs(cls):
@@ -41,9 +41,19 @@ def to_lib(self, src):
     def sources(cls, interpreter):
         for src in super(PyPy3Posix, cls).sources(interpreter):
             yield src
+        # Also copy/symlink anything under prefix/lib, which, for "portable"
+        # PyPy builds, includes the tk,tcl runtime and a number of shared
+        # objects. In distro-specific builds or on conda this should be empty
+        # (on PyPy3.8+ it will, like on CPython, hold the stdlib).
         host_lib = Path(interpreter.system_prefix) / "lib"
+        stdlib = Path(interpreter.system_stdlib)
         if host_lib.exists() and host_lib.is_dir():
             for path in host_lib.iterdir():
+                if stdlib == path:
+                    # For PyPy3.8+ the stdlib lives in lib/pypy3.8
+                    # We need to avoid creating a symlink to it since that
+                    # will defeat the purpose of a virtualenv
+                    continue
                 yield PathRefToDest(path, dest=cls.to_lib)