If the backup files contain the date specification in their name, it would be simple to cycle over the past 10 days date, that is every day you have something like
to_delete=$(date -d "10 days ago" +%Y%m%d)
and then use this date in the FTP delete statement
or whatever be the name of your archives. In this way you can manually delete the files older than 11 days and then start to delete the 10-days old file every day.
An alternative method would be to check the timestamp of the remote files. To do this you can try the FTP modtime
command or even better you may try lftp as client software, that permits some coding (e.g. if/then constructs) and a finer control over the task to accomplish. HTH.