Something similar, but not quite identical, to what you want is true.
It is not possible to guarantee strategies which ensure that the game continues forever, if neither has a winning strategy. For example, consider the really sad game whose only moves for either player are "pass" or "resign." Neither player has a winning strategy - all they can do is refrain from resigning - but neither player can ensure that the game goes on forever, either (since each player could resign at any given moment).
However, what we can guarantee is that at least one player has a non-losing strategy (i.e. when they play according to that strategy, either they win or the game continues forever). This follows from the Gale-Stewart theorem; this is a significant strengthening of Zermelo's theorem, and represented the first real step into the land of infinite games.
Call your game $G$. Consider the related game $G^*$ where players $a$ and $b$ play the game you've described, and if it goes on forever then player $b$ is said to win. This is an example of an open game, and the Gale-Stewart theorem says that every open game is determined. A winning strategy for $a$ in $G^*$ is also a winning strategy for $a$ in $G$; meanwhile, a winning strategy for $b$ in $G^*$ is a non-losing strategy for $b$ in $G$. So we see that one player or the other has a non-losing strategy in your game $G$, as desired.
So in fact we have the following "lopsided" determinacy principle:
In a (perfect-information, $2$-player) no-draws-possible but not-necessarily-terminating game $G$, either player $a$ has a winning strategy or player $b$ has a non-losing strategy.
There are essentially two proofs of the Gale-Stewart theorem. One is a "ranking" argument, where we assign nodes in the "game tree" ordinals (or "$\infty$") in a recursive way. The other "magical" one is the following: looking at the game tree, color a node red if player $a$ has a winning strategy from that node and blue otherwise. Now show that if the root of the game tree (= the beginning of the game) gets colored blue, then $b$ has a non-losing strategy. HINT: think about what color the nodes directly above a given blue node can get, based on the parity of that node (even length or odd length).
There are much, much stronger determinacy principles out there - see in particular Borel determinacy, due to Martin. The general study of determinacy principles is a major topic in set theory and descriptive set theory, and has surprising connections to large cardinals and "regularity" properties (like measurability) of sets of real numbers. See this essay of Larson.