After confirmation that any democratic republic qualifies, I'd say the Democratic Republic of the Congo provides a recent example. While the DRC is not a democracy by most standards, it's an interesting example of a country that presents itself as a democratic republic.
In 2016, the BBC reported:
More than 20 people have been killed in clashes between protesters and security forces in the Democratic Republic of Congo's capital, Kinshasa, over President Joseph Kabila's failure to give up power, a UN official has said.
Some of the dead were shot at close range by troops, witnesses said.
Mr Kabila's 15-year rule was due to have ended on Monday at midnight, but has been extended to 2018.
Mr Kabila's main rival said the refusal to give up power amounted to a coup.
Wikipedia provides a timeline on the matter:
Kabila's second term as president of the Democratic Republic of Congo was due to end on 20 December 2016. A statement issued by his spokesperson on 19 December 2016, stated that Joseph Kabila would remain in post until a new president is in place following elections which will not be held until at least April 2018. Kabila subsequently installed a new cabinet led by prime minister Samy Badibanga, resulting in protests in which at least 40 people were killed. Under articles 75 and 76 of the Constitution of the Democratic Republic of Congo, should the office of the president become vacant, the Chairman of the Senate, presently Léon Kengo would assume the presidency in an acting capacity.
On 23 December, an agreement was proposed between the main opposition group and the Kabila government under which the latter agreed not to alter the constitution and to leave office before the end of 2017. Under the agreement opposition leader Étienne Tshisekedi will oversee that the deal is implemented and the country's Prime Minister will be appointed by the opposition.
In late February 2018 the ministry of international affairs of Botswana told Kabila that it was time to go and said the "worsening humanitarian situation" in DRC is compounded by the fact that "its leader has persistently delayed holding elections, and has lost control over the security of his country".
On December 30, 2018 the presidential election to determine the successor to Kabila was held. Kabila endorsed Emmanuel Ramazani Shadary, his former interior minister. On January 10, 2019, the electoral commission announced opposition candidate Félix Tshisekedi as the winner of the vote.
As for more stable democracies (e.g. going by the Economist Intelligence Unit's Democracy Index), it will be very hard to find examples of leaders overstaying their position.
That's because being a full democracy requires having solid institutions which prevent its leaders from successfully pulling off the things described in the question.
To qualify that, I'll cite the three highest ranking classification definitions from Wikipedia:
Full democracies are nations where civil liberties and fundamental political freedoms are not only respected but also reinforced by a political culture conducive to the thriving of democratic principles. These nations have a valid system of governmental checks and balances, an independent judiciary whose decisions are enforced, governments that function adequately, and diverse and independent media. These nations have only limited problems in democratic functioning.
Flawed democracies are nations where elections are fair and free and basic civil liberties are honoured but may have issues (e.g. media freedom infringement and minor suppression of political opposition and critics). These nations have significant faults in other democratic aspects, including underdeveloped political culture, low levels of participation in politics, and issues in the functioning of governance.
Hybrid regimes are nations with regular electoral frauds, preventing them from being fair and free democracies. These nations commonly have governments that apply pressure on political opposition, non-independent judiciaries, widespread corruption, harassment and pressure placed on the media, anaemic rule of law, and more pronounced faults than flawed democracies in the realms of underdeveloped political culture, low levels of participation in politics, and issues in the functioning of governance.
The DRC example I cited doesn't fit any of the above classifications. In fact, the DRC ranks second to last (before North Korea) as an authoritarian regime.
For context, the United States ranks 25th as a flawed democracy. The top 5 full democracies is made up of Scandinavian countries (Norway, Iceland, Sweden, and Finland) and New Zealand.
