Are all Klondike solitaire games winnable?

No. Klondike is the game most people picture when they hear the word solitaire, and it is not built to be beatable every time. A solver that can see every card, including the face-down ones, wins about 90.5% of turn-1 deals and about 81.9% of turn-3 deals. That is the theoretical best: winnable in principle by a player with perfect information.

In a real game you cannot see the face-down cards, so your win rate sits well below that number, and one bad early move can quietly ruin a deal. Turn-3 is harder than turn-1 because drawing three cards at a time reaches fewer of them. The deals that remain have no winning line at all.