Gomoku on an infinite big board I always used to play Gomoku in school on paper, and if we reached the edge of the field, we just put another one at that side.
And now I just saw that black can always win on 1 15x15 board. But what is about the way I used to play it? Is black also winning there, if both players play perfectly?

There is discussion of this topic in volume 3 of Winning Ways (Berlekamp et al. 2003, pp 740–741). They discuss the general case of $n$-in-a-row on infinite boards.  A strategy-stealing argument shows that each game is either a draw or a first-player win.  It should be clear that if $n$-in-a-row is a first-player win, then so too is $m$-in-a-row for $m<n$, so the only real question is what is the largest $m$ that is a first-player win. They provide a proof that 9-in-a-row is a draw, and say  “T.G.L. Zetters… recently showed that the second player can even draw 8-in-a-row”.
