If a player draws their last card do they lose immediately or at the next state-based action check?

Drawing a card never itself causes a player to lose. Instead, an empty library creates a condition that is evaluated when State-Based Actions (SBAs) are checked. CR 704.3 specifies that SBAs are checked and performed as a last step before any player would receive priority. Until that check happens, the game continues.

The specific SBA in question is CR 704.5b: 'If a player attempted to draw a card from a library with no cards in it, that player loses the game.' Note the trigger is the attempt to draw from an empty library, not merely having an empty library. So if a player draws their very last card, they have successfully drawn it — their library is now empty but no draw-from-empty has occurred yet.

The loss will be applied at the very next SBA check, which occurs before any player receives priority (e.g., before spells or abilities can be cast, before combat steps advance, etc.). In practice this usually feels nearly instantaneous, but there is a tiny window where the game state exists with an empty library and no loss yet applied.

Concrete example: Alice has one card left in her library and draws it during her draw step. She now holds that card in hand and her library is empty. Before she receives priority to take any action in her main phase, SBAs are checked and she loses the game under CR 704.5b — she never gets to play that last drawn card.

This distinction matters in edge cases: if something grants a player an extra card draw in response to the draw (via a replacement effect or a trigger that fires and resolves before the next SBA check), the loss may occur sooner or be modified. But purely from the base rule, the loss is applied at the SBA check, not mid-draw.