пробуем так

.drone.yml

@@ -11,11 +11,11 @@ steps:
       - pytest tests/ -v || true
 
   - name: build-image
-    image: docker:latest
-    volumes:
-      - name: docker-socket
-        path: /var/run
+    image: docker:dind
+    privileged: true
     commands:
+      - dockerd &
+      - sleep 5
       - docker build -t gptchat:latest .
     when:
       branch:
@@ -24,11 +24,11 @@ steps:
         - push
 
   - name: deploy
-    image: docker:latest
-    volumes:
-      - name: docker-socket
-        path: /var/run
+    image: docker:dind
+    privileged: true
     commands:
+      - dockerd &
+      - sleep 5
       - docker stop gptchat || true
       - docker rm gptchat || true
       - docker run -d \
@@ -42,8 +42,3 @@ steps:
         - main
       event:
         - push
-
-volumes:
-  - name: docker-socket
-    host:
-      path: /var/run/docker.sock