Maybe this?

.drone.yml

@@ -10,19 +10,18 @@ steps:
   image: golang:1.16.4-alpine
   volumes:
     - name: build
-      path: /build
+      path: /drone/src/build
   environment:
     CGO_ENABLED: 0
     GOOS: linux
   commands:
-    - go build -a -installsuffix cgo -o /build/app ./discord
-    - ls /build
+    - go build -a -installsuffix cgo -o ./build/app ./discord
 - name: build-publish-image
   image: plugins/docker
   privileged: true
   volumes:
     - name: build
-      path: /build
+      path: /drone/src/build
   settings:
     cache_from:
       - "dutchellie/pepebot:latest"
@@ -52,20 +51,19 @@ steps:
   image: golang:1.17-alpine
   volumes:
     - name: build
-      path: /build
+      path: /drone/src/pepeservice/build
   environment:
     CGO_ENABLED: 0
     GOOS: linux
   commands:
     - cd ./pepeservice
-    - go build -a -installsuffix cgo -o /build/app .
-    - ls /build
+    - go build -a -installsuffix cgo -o ./build/app .
 - name: build-service
   image: plugins/docker
   privileged: true
   volumes:
     - name: build
-      path: /build
+      path: /drone/src/pepeservice/build
   settings:
     cache_from:
       - "dutchellie/pepebot_service:latest"

Dockerfile

@@ -8,9 +8,6 @@
 
 FROM alpine:latest  
 RUN apk --no-cache add ca-certificates
-WORKDIR /build
-RUN ls
-RUN ls /build
-COPY . /root/
+COPY ./build/app /root
 WORKDIR /root
 CMD ["./app"]  

pepeservice/Dockerfile

@@ -9,9 +9,6 @@
 
 FROM alpine:latest
 RUN apk --no-cache add ca-certificates
-WORKDIR /build
-RUN ls
-RUN ls /build
-COPY . /root/
+COPY ./build/app /root
 WORKDIR /root
 CMD ["./app"]