For everyone who might be interested in a not-so-quick fix to keep an older Armbian install running, here is what you can do until base-files become available for armhf:
Download the latest base-files version of amd64 and armhf.
Unpack with dpkg-deb -x base-files-xx.xx.xx_long_text_armhf.deb armhf and dpkg-deb -x base-files-yy.yy.yy_long_text_amd64.deb armhf_new in two directories armhf and armhf_new.
Use diff -qr armhf armhf_new to test for differences. The file arm*/usr/bin/locale-check is different because different architectures, but the creation time shows that the content is the same. Copy it over: cp -a armhf{_new}/usr/bin/locale-check
The lib64 symbolic link at the root of armhf_new and its target are unnecessary: rmdir $(readlink -f armhf_new/lib64) && rm armhf_new lib64 .
Use dpkg-deb --control base-files-yy.yy.yy_long_text_amd64.deb armhf_new/DEBIAN to have the control files as well.
edit armhf_new/DEBIAN/control to have line 4 read armhf instead of amd64. Remove the `lib64` words from /armhf_new/DEBIAN/*. Use grep -nrw armhf_new/DEBIAN -e lib64 to find them.
Fix the md5sum with md5sum armhf_new/usr/bin/locale-check and edit armhf_new/DEBIAN/md5sums accordingly.
Fix the ownership of the files and directories with sudo chown -R root:root armhf_new/*
Build the new deb package: sudo dpkg-deb -b armhf_new base-files_yy.yy.yy.deb and install it: sudo dpkg -i base-files_yy.yy.yy.deb
Upgrade formerly stuck packages like armbian-bsp-cli-odroidxu4-current: sudo apt update && sudo apt full-upgrade
Clean up.
It would really be easier to have an up-to-date version of base-files for armhf, but this has happened twice for me and at least once for another guy on the forum, so I hope the fix can be of help in the future.