The answer
Normally no. The battery powers the device but the relevant secret is retained in non-volatile storage; access still depends on the device and backup.
What to do next
Avoid opening a swollen battery device; use a compatible replacement and independently test the recovery.
Trezor hardware wallets
Hardware wallet related to device/backup security; not a guarantee of recovery or anonymity. Availability, supported assets and local eligibility must be checked before signing up or buying.
Visit Trezor hardware wallets ↗Official provider link · Compare suitability and availability before choosing.