For a $2$ by $2$ board, Natalia wins on the second move clearly. For an $n$ by $n$ board, Ana must move towards Natalia on the first turn. Then Natalia mirrors Ana's move such that Ana and Natalia are on opposite vertices of a $n-1$ by $n-1$ sub-board. If Ana attempts to move away from Natalia to break out of the "sub-board," Natalia moves in the same direction as Ana, preserving the sub-board. Eventually Ana must move towards Natalia again, allowing the latter to make the sub-board smaller again, and since it is always Ana's turn to move when Natalia forms the sub-board, Natalia can eventually form a $2$ by $2$ sub-board, where Ana loses. Thus Natalia always wins.