Hello,
It is possible only if your backend servers shares user sessions.
For example :
1) user 1 connects for the first time to server 1, server 1 returns a session cookie and synchronize cookie with server 2.
When server 1 fall down, the F5 will monitor this and send the user 1 on server 2
2) user 1 connects to server 2, if server 2 has received session information from server 1 => it can continue serving user 1 in the same way. If server 2 didn't receive any info about user 1 session from server 1 it restarts from zero...
So it is more a backend feature not the role of the F5 load balancer.
Hope it helps
Regards