Add GitHub action to deploy to gh-pages

.github/workflows/gh-pages.yml

@@ -0,0 +1,35 @@
+name: Deploy to gh-pages
+
+on:
+  push:
+    branches: [main]
+
+jobs:
+  build:
+    runs-on: ubuntu-latest
+
+    steps:
+      - uses: actions/checkout@v4
+
+      - uses: pnpm/action-setup@v4
+        name: Install pnpm
+        with:
+          run_install: false
+
+      - name: Install Node.js
+        uses: actions/setup-node@v4
+        with:
+          node-version: 20
+          cache: "pnpm"
+
+      - name: Install dependencies
+        run: pnpm install
+
+      - name: Build doc+demo site
+        run: pnpm --filter starlight-client-mermaid-docs build
+
+      - name: Deploy 🚀
+        uses: JamesIves/github-pages-deploy-action@v4
+        with:
+          branch: gh-pages
+          folder: docs/dist # only deploy the b