@Miniman has answered this correctly but there still seems to be some confusion.
What an assassin can do (PHB p.97):
You have advantage on attack rolls against any creature that hasn’t taken a turn
in the combat yet. In addition, any hit you score against a creature that is surprised is a critical hit.
What surprised is (PHB p.189):
Any character or monster that doesn’t notice a threat is surprised at the start of the encounter.
If you’re surprised, you can’t move or take an action on your first turn of the combat, and you can’t take a reaction until that turn ends.
So surprise has a start "the start of the encounter" and the effects of being surprised ("you can’t move or take an action on your first turn of the combat, and you can’t take a reaction until that turn ends") have a definite end - the end of your first turn.
It is nonsensical to claim that being "surprised" is a situation that can outlast the end of its effects; can being "unconscious" outlast the effects of unconsciousness?
Following the sequence of play on p.189 and using a very simple 2 creature example:
- Determine surprise: Alice the Assassin is not surprised; Victor the Victim is surprised.
- Establish positions: we will take that as read.
- Roll initiative: There are 2 possibilities:
- Alice rolls higher then Victor
- Victor rolls higher than Alice
If Alice rolled higher then Victor
Alice goes first and can:
a) move out of Victor's reach secure in the knowledge that Victorwill not get an Attack of Opportunity - this is a reaction and Victor does not get them until after his first turn.
b) use her assassin ability to i) attack with advantage because Victor has not taken a turn (this would be true without surprise and ii) get a critical on any hit because of surprise. If Alice has additional attacks and any bonus actions that grant attacks (e.g. Flurry of Blows or Two Weapon fighting) then this will apply to all of them.
Victor then takes his turn and "recovers from surprise".
If Victor rolled higher than Alice
- Victor takes his turn and "recovers from surprise"
- Alice takes her turn and
a) would allow Victor an attack of opportunity if she moves out of his reach - he can now take reactions
b) cannot use either of her assassin abilities because: for i) Victor has had his turn and for ii) he is no longer surprised.
Begin the next round
If Alice won the initiative then this combat is likely over. If Alice lost the initiative the only advantage she got from surprising Victor was not having him act in the first turn.