However, she did not directly answer a question about her plans beyond the current five-year Holyrood term.
Elsewhere, Ms Sturgeon said the Scottish Government intends to take steps towards a second independence referendum next year.
There has been widespread speculation about Ms Sturgeon’s plans for life after politics, fuelled by a recent interview in which she spoke of potentially becoming a foster mother.
Scottish Conservative leader Douglas Ross previously predicted the First Minister would resign before the 2026 Holyrood election.
In an interview with BBC political editor Laura Kuenssberg, Ms Sturgeon was asked if she could guarantee she will lead the party at the next election.
Ms Sturgeon said: “I will fulfil the mandate I have been given to govern as First Minister for this term of the Scottish Parliament.”
Pushed on her plans beyond this, she said: "It is almost as if my opponents have concluded they can't beat me or remove me from office themselves, so they're kind of crossing their fingers and hoping that I'll remove myself from office.
"But they are going to be really disappointed because I'm going to be around a lot longer."
She added: "I was elected seven months ago, having asked people in Scotland to put their trust in me for a five-year term as First Minister.
"They gave me that trust and they re-elected me. We face serious times as a country and I intend to fulfil that mandate."
Ms Sturgeon told the BBC: "I’m 51 years old. I have no intentions of going anywhere right now as First Minister, but I hope I’ll still be relatively young when I get to the point of contemplating other things.”
Scottish Conservative chief whip Stephen Kerr said: “Nicola Sturgeon has confirmed she is going to spend the next four years obsessing about breaking up the UK and failing Scotland, just as she has done for the last 14 years.”
Meanwhile, Ms Sturgeon said the Scottish Government intended “to take the steps that enable a referendum to happen before the end of 2023” next year.
She added: “But equally importantly, and perhaps more importantly, what will also happen and step up in earnest next year is the setting out of the substantive case for Scotland becoming independent.
"That’s work that has begun again in the Scottish Government, and we will set that out – the advantages and opportunities of independence, but also how we will navigate some of the more difficult issues.”
Ms Sturgeon wants a second referendum to be held before the end of 2023, Covid permitting.
However, she previously paused work on independence due to the pandemic and has faced internal criticism over a perceived lack of urgency.
The First Minister said she hoped Scotland would be out of the "acute phase” of the pandemic “as we get into the early part of next year”.
She said she wanted a referendum to be held when people “aren’t living their life under restrictions, when we don’t still have hospital wards full of Covid patients, when we’re not still in the acute phase of a pandemic”.