The answer
Sometimes. Paymasters can sponsor selected UserOperations, but sponsorship depends on policy, supported chain and available deposits. If declined, the wallet may need another fee route.
What to do next
Read the specific paymaster rejection and use a supported funding or sponsorship option.
ethereum.org — Wallet and account documentation
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Ledger hardware wallet
Relevant hardware-backed signing for supported assets; it does not eliminate gas fees, contract vulnerabilities or approval risks.
Explore Ledger hardware wallet ↗Official website; affiliate programme available to apply for. No tracked link or partner approval yet.