I have doubts about whether my solution is right.
Let $x>z>y>1$. Given the extended game above, we have two subgames and will find the Nash equilibria of these.
Observe that player 2 will play $b$ in the right subgame. In the left subgame player 2 will be indifferent between playing $a$ and $b$. Let $0 \le r \le 1$ such that $(r,1-r)$ is a mixed strategy for player 2 and such that the player plays this strategy in the left subgame.
Player 1 has expected payoffs $$ E u(s) = (rx+(1-r)y) \cdot 1_{s=A} + z \cdot 1_{s=B} $$ Observe that player 1's best response is $A$ if $r \ge (z-1)/(x-y)$ and $B$ if $r \le (z-1)/(x-y)$, hence SPNE will be $$ \{(A,(r,1-r))|r \ge (z-1)/(x-y) \} \cup \{(B,(r,1-r))|r \le (z-1)/(x-y)\} $$ Is this correct?