This answer suggests the answer is "yes", but you need the intervention of the arbiter, "no increment" is a hard requirement, and you must not be playing blitz. You must have less than two minutes left, and you must be willing to accept a draw (i.e. you cannot say "I have at least a draw, but might still win").
From the FIDE Rules of Chess:
Guidelines III. Games without increment including Quickplay Finishes
III.1 A ‘quickplay finish’ is the phase of a game when all the
remaining moves must be completed in a finite time.
III.2.1 The Guidelines below concerning the final period of the game
including Quickplay Finishes, shall only be used at an event if their
use has been announced beforehand.
III.2.2 These Guidelines shall apply only to standard chess and rapid
chess games without increment and not to blitz games.
...
III.4 If the player having the move has less than two minutes left on
his clock, he may request that an increment extra five seconds be
introduced for both players. This constitutes the offer of a draw. If
the offer refused, and the arbiter agrees to the request, the clocks
shall then be set with the extra time; the opponent shall be awarded
two extra minutes and the game shall continue.
III.5 If Article III.4 does not apply and the player having the move
has less than two minutes left on his clock, he may claim a draw
before his flag falls. He shall summon the arbiter and may stop the
chessclock (see Article 6.12.2). He may claim on the basis that his
opponent cannot win by normal means, and/or that his opponent has been
making no effort to win by normal means:
III.5.1 If the arbiter agrees that the opponent cannot win by normal
means, or that the opponent has been making no effort to win the game
by normal means, he shall declare the game drawn. Otherwise he shall
postpone his decision or reject the claim.
III.5.2 If the arbiter postpones his decision, the opponent may be
awarded two extra minutes and the game shall continue, if possible, in
the presence of an arbiter. The arbiter shall declare the final result
later in the game or as soon as possible after the flag of either
player has fallen. He shall declare the game drawn if he agrees that
the opponent of the player whose flag has fallen cannot win by normal
means, or that he was not making sufficient attempts to win by normal
means.
Since you can prove a draw, you can argue that the opponent cannot win by normal means, invoking III.5.
In a situation like this, the player got into a losing position with a glimmer of hope for a draw, but they've also used too much time to be able to pull that off. They misplayed both on the board and the clock, and the loss is a natural and fair consequence.
– Nelson O Jan 16 '24 at 19:59