Backup Error with Msg 3201, Level 16, State 1, Line 1 & Operating system error 3
SQL Server can only see network drives that are mapped in the Windows Profile of the SQL Server service account. So you'd have to log on as that account (not your own) and map the drive, for SQL Server to be able to use it for backups. Backups run using the credentials of the SQL Server service account, not yours. As suggested in the comment above, try using a full UNC patch instead (e.g. '\fileserver\fileshare\backups\backupfile.bak' which will not require mapped drives.

Note that the SQL Server service account will require read/write permissions to the destination. Being able to access external file system is one very good reason to use a domain account for your SQL Server service account, rather than LOCAL SYSTEM etc..