Is it possible to play against a friend (by random selection) in Ranked Mode?

There is no sure way to face a pre-determined opponent in ranked mode. That's the simple answer.

However, it is not impossible to do so, just generally quite unlikely. There are more players at lower ranks than at higher ranks, so it is less likely to happen at low ranks and likelihood increases at higher ranks.

Anecdotally, I have accomplished this in Arena mode once. My brother (RealID friend on Battle.net) and I had both drafted arena decks. We would time our entry in the queue to be as close as possible. After a few tries, we actually did get matched against each other, but weren't able to do so again. It really just comes down to a bit of timing and a lot of luck.

You can play against your friends in Ranked in hearthstone - this is shown with streamers play people on their friends list and end up chatting over the Battle.net chat during the game.

However, when you're not in legend/near legend, it's much harder to get paired vs a friend, as there are more people in that band.