Does a flipped from hidden pyke survive when opponent plays the ball to a Battlefield

It depends on which Pyke you flipped and whether you paid the [C] — The Ball's "Deal 2 to enemy units here" chooses nothing, it just hits every enemy unit at its location, so it comes down to Might. Pyke, Dockside Butcher at his base 2 [M] dies to it (2 damage ≥ 2 Might); Dockside Butcher who paid the [C] additional cost is 4 [M] this turn and survives with 2 damage marked; Pyke, Returned (3 [M]) survives with 1 Might's worth of room left.

The opponent can legally play The Ball straight onto your occupied battlefield — Deploy grants gear permission to be played to any battlefield, including enemy-occupied ones (Rule 830.1.b.1) — and because the damage isn't targeted, Deflect/Not So Fast can't stop it, and Backline is irrelevant (it only modifies combat damage assignment).

One timing note if you're hoping to keep a fresh Dockside Butcher alive: The Ball has no Reaction, so it can't be played during the Closed State that Pyke's own "when you play me" trigger creates — the +2 [M] will already be applied before The Ball can be played that turn. But if The Ball arrives on a later turn, the +2 has expired, Pyke is back to 2 [M], and he dies (Rule 142.4.b).