There were two solutions:
- Enlarge disk C: on Azure VM... the complicated way (stop VM, dismount, enlarge the VHD, mount, recreate the VM etc...)
- Compress some unused stuff on disk... the easy way
So I've chosen the easy way. I installed Folder Size 64bit, the expansion for Windows Explorer, and found that one of the biggest folders is "%WINDIR%\WinSxS". It shouldn't be deleted as there are some useful installation files, but can be compressed. So I did it as described here: http://dandar3.blogspot.com/2013/01/how-to-ntfs-compress-windows-winsxs.html.
Compression took 30 minutes. After that I've 3,90GB free space on C: drive, enough for few next months.