Germany’s anti-immigration Alternative for Germany (AfD) is celebrating a “historic success”, with the far-right party on course for a big victory in the eastern state of Thuringia.

The AfD is set to win almost a third of the vote, according to a projection for public broadcaster ARD, nine points ahead of the conservative CDU, and far in front of Germany’s three governing parties.

The result would give the far right its first vote win in a state parliament since World War Two, although it has little hope of forming a government in Thuringia.

The AfD came a close second in Sunday’s other big state election, in the more populous neighbouring state of Saxony.

Projections there gave the CDU almost 32% of the vote, a point ahead of the AfD, again far ahead of the three parties running the national government – the Social Democrats, Greens and liberal FDP.

The AfD’s top candidate in Thuringia, Björn Höcke, who is a highly controversial figure in Germany, hailed a “historic victory” and spoke of his great pride. He reportedly failed to win a direct mandate for the state parliament, but could still win a seat because he is top of his party list.

Mr Höcke’s party has been designated as right-wing extremist and he has been fined for using a Nazi slogan, although the former history teacher denies knowingly doing so.

One of Germany’s best-known Holocaust survivors, Charlotte Knobloch, pointed out that the election had taken place 85 years to the day since the outbreak of World War Two. The result had left the country in danger of becoming “more unstable, colder and poorer, less safe and less worth living in”, she said.

With federal elections only a year away, the AfD is second in national opinion polls. Co-leader Alice Weidel said the result was a “requiem” for the three parties running Germany. And it was clear that voters in both eastern states wanted her party in government: “Without us a stable government is no longer possible at all.”

That message was repeated by Björn Höcke, who suggested there were plenty of CDU voters who would be happy if they worked together.

Without the support of other parties, the AfD cannot govern in Thuringia, and the CDU has made clear it will not consider ruling with the far right.

Mathematically, the conservatives will need support from parties on the left to form a majority.

Some five million Germans in the east were eligible to vote on Sunday and, according to a survey for public broadcaster ZDF, 36% of under-30s in Thuringia voted for the AfD, far more than any other party.

The biggest issue for AfD voters on Sunday was immigration, and in particular the issue of refugees and asylum.

“Politicians have promised a lot, particularly concerning migration and foreigners,” AfD voter Michael told the BBC in Thuringia’s state capital, Erfurt.

“But nothing happened. Nothing. Just promises came from these parties. Now I have my party. And I stand with my decision,” he said, standing beside his partner Manuela, who agreed that people wanted change.

The asylum issue was re-ignited nationally little more than a week before the vote, when three people were murdered at a street festival at Solingen in western Germany, and a Syrian man facing deportation was arrested on suspicion of carrying out the attack.

AfD deputy leader Beatrix von Storch told the BBC’s Newshour programme that political opponents had been attacking her party’s asylum policies as extremist for years. “Two days ahead of the election they started to do what we always said had to be done,” she said, referring to a series of government measures aimed at toughening asylum laws.

The AfD also wants to stop weapons supplies to Ukraine, as does a new party heading for third place in both states, left-wing populist leader Sahra Wagenknecht’s BSW.

Although she has similar ideas to the AfD on Ukraine, Ms Wagenknecht has, like the other parties, refused to take part in any coalition with the far right.

If the projections are confirmed, the AfD is on course to win 32 seats in the 88-seat Thuringia state parliament, and the CDU 23 seats, with only one of the three parties in the national government represented.

That would give the AfD more than a third of the seats, handing it a blocking minority on decisions that require a two-thirds majority, including changes to the state constitution or appointing judges.

Chancellor Olaf Scholz’s Social Democratic Party (SPD) is set to win just six seats, with none for the Greens and liberal FDP.

In Saxony, the conservatives are on course to win 42 seats, just ahead of the AfD with 40, while Sahra Wagenknecht’s party is in third with 15 seats.

Sunday’s elections have underlined the unpopularity of Germany’s ruling “traffic-light” coalition, so named because of the red, yellow and green of the party colours.

A third eastern state, Brandenburg, is due to vote in three weeks’ time and although the AfD is ahead in the opinion polls, the Social Democrats and conservatives are only a few points behind.

While Björn Höcke hailed his party’s victory with supporters in Erfurt, anti-AfD protesters gathered outside the Thuringia state parliament.

The AfD has been classified as right-wing extremist by domestic intelligence in Thuringia as well as Saxony. In May, a German court ruled that the BfV intelligence agency was justified in placing the AfD under observation for suspected extremism.

Among the protesters was Hannah, a local student, who said she was very worried by the result: “I think there are a lot of people who are aware they have Nazi policies and don’t care. Germany has some kind of responsibility on that matter.”

The rise of Sahra Wagenknecht’s populist party had a direct impact on the Left party, which won the last election in Thuringia but has now slipped into fourth place.

Bodo Ramelow. the Left-party state premier of Thuringia, who had led a coalition with the SPD and Greens, said the election campaign had been characterised by fear and that he was “fighting against the normalisation of fascism”.