Home Office data shows 237 such calls were made to the South Yorkshire Fire and Rescue Service in the year to March – 14 more than the year before.
They included 68 calls to help animals who found themselves in trouble and 169 callouts to remove objects from people.
Across England the number of times firefighters were drafted in to help animals increased from 4,724 to 5,159 over the year.
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The most common reason was to help a trapped pet, which was quoted in a fifth of incidents attended nationally, closely followed by rescuing pets from a height.
Animal charity the RSPCA said its staff rescues tens of thousands of stuck or trapped animals every year.
Steve Bennett, deputy chief inspectorate officer, said: "While our staff are trained and equipped to help a lot of them out of trouble, there may be some situations where they need a helping hand to ensure both the rescuers and the animals remain safe.
"We can request the assistance of the fire and rescue service just like any member of the public can do, and we're incredibly grateful to crews across England and Wales who are animal-lovers just like us and will always lend a hand to help an animal in need if they're available."
The number of people requiring help with stuck objects also increased nationally from 5,311 to 5,632.
Two-thirds of incidents saw someone needing help removing a ring, while trapped limbs accounted for 16% of calls.
Overall, South Yorkshire firefighters attended 12,246 incidents in 2020-21 – including 6,067 fires – down from 13,860 the year before.
They included 2,566 non-fire related incidents, which may be related to flooding, assisting people trapped in lifts and road traffic accidents.
The National Fire Chief's Council chairman Mark Hardingham said: “Despite the huge amount of positive and proactive work carried out nationally and locally, incidents, and sometimes very serious incidents, do still happen.
"It is of critical importance that we maintain a well-resourced fire and rescue service to respond professionally and safely to national and local emergencies."