Can a player activate a mana ability during another player's turn without being given priority?

Mana abilities are a special category of activated (or triggered) abilities that produce mana as part of their effect and do not use the stack. Because they don't use the stack, the normal priority system does not apply to them — a player does not need to hold priority to activate a mana ability (CR 605.3a).

The key rule is CR 605.3b: a player may activate a mana ability whenever they have priority, whenever they are casting a spell or activating an ability that requires mana payment, or whenever a rule or effect asks them to pay mana. This means even during another player's turn, if you are required or permitted to pay mana, you can activate your mana abilities freely without being formally passed priority.

For example, if an opponent casts a spell with a triggered ability that says 'target player pays 2 mana or loses 2 life,' you — the targeted player — may tap your lands to produce mana to pay that cost during your opponent's turn, even though you were never formally given priority in the usual sense.

It is important to note that this only applies to mana abilities. Other activated abilities (even those that incidentally involve mana) that use the stack still require priority. An ability is a mana ability only if it meets the definition in CR 605.1: it is an activated or triggered ability that could put mana into a player's mana pool and doesn't have a target or a planeswalker loyalty cost.