That is correct, the vampire will die. Combat damage is dealt simultaneously if none/all of the creatures have First Strike/Double Strike (if two creatures were to fight Contested Cliffs, the situation would be the same). After damage is assigned, it is dealt simultaneously, then triggered abilities are placed on the stack. This happens just before a player receives priority, which means that you have to check for state-based actions, one of which is to place creatures with lethal damage in the graveyard.
510.1. First, [...] each creature assigns its combat damage
510.2. Second, all combat damage that's been assigned is dealt simultaneously. This turn-based action doesn't use the stack. No player has the chance to cast spells or activate abilities between the time combat damage is assigned and the time it's dealt. This is a change from previous rules.
510.3. Third, any abilities that triggered on damage being assigned or dealt go on the stack. (See rule 603, "Handling Triggered Abilities.")
510.4. Fourth, the active player gets priority. Players may cast spells and activate abilities.
704.5g If a creature has toughness greater than 0, and the total damage marked on it is greater than or equal to its toughness, that creature has been dealt lethal damage and is destroyed. Regeneration can replace this event.