The defunct Ciudad Real Airport, also marketed as "South Madrid Airport", is/was about 200 km (227km from the Puerta del Sol, the official distance) away from Madrid

It just fails to beat your 'Frankfurt' Hahn, but Lydd Airport is also called London Ashford Airport, even though it's 117 km from central London. Given usual traffic conditions, that's probably a longer drive with a car than from Hahn to Frankfurt.

It's a bit of a cheat, but Bozeman Yellowstone International Airport (BZN) is approximately 150 km from the geographic center of Yellowstone National Park. Of course, Yellowstone National Park is huge; the closest point in the park is "only" about 75 km away as the crow flies, while the farthest point is about 200 km away. The park headquarters are 96 km from BZN.

Similarly, as pointed out by @user71659, Fresno Yosemite International Airport (FAT) is approximately 105 km as the crow flies from Yosemite Village. This is perhaps a bit less of a cheat than the Yellowstone case, since while there is not a town called "Yellowstone" in Yellowstone National Park, there is a town called "Yosemite Village" in Yosemite. (This does stretch the definition of a "city", of course.)

I'm going to nominate Anadyr airport in Russia's far North-East.

Now, it's only 10 km from the town, but it is on a different side of the bay and there are no roads between airport and the town it serves.

Not a problem in winter, since the bay will freeze and allow driving over. But between seasons, I think it's not uncommon when you can't reach airport from town or vice versa for days or weeks, since the weather is brutal. There is an option to commute to airport with helicopter, which will again depend on weather.

Not going to win, but another that’s over 100km from the city with the same name is Stockholm Skavsta Airport which is 106km by road to the Stockholm city center.

Not going to win, but the Lhasa Gonggar Airport (LXA) is 97km away from downtown Lhasa. Nevertheless, it is going to win in one aspect: it is the only civilian airport serving Lhasa, making Lhasa the most distant city to any airport among all the cities that have been mentioned.

A new airport is under construction in Lhasa to allow easier access to the city center...and the new airport is 70 km away from downtown Lhasa. The main problem here is that Tibet is too mountainous and it is hard to find a large enough expanse of flat land suitable for an airport.

The distance

According to Google Maps, the distance from Bangkok to U-Tapao is upward of 170 kilometres by road, as shown in the screenshot below. The direct distance, as pointed out in a statement by Eva Air (in the quote below, in bold) is about 140 kilometres.

The future

There are plans for U-tapao Rayong-Pattaya Airport to become Bangkok's third airport. According to The Blue Swan Daily:

The Royal Thai Navy, operator of Utapao Rayong-Pattaya International airport, recently unsealed tender bids for the USD9 billion redevelopment of the facility under the Thai Government’s Eastern Economic Corridor (EEC) scheme.

Its summary of the article lists:

Utapao will ultimately take its place as a ‘Bangkok’ airport, but one dedicated to tourists as it serves nearby Pattaya.

The past

It's not completely unprecedented for flights to Bangkok to head to U-Tapao. In 2008, when both of Bangkok's main airports were closed, some airlines chose to direct their flight there. From web.archive.org's capture of the Eva Air website (emphasis is mine):

Suvarnabhumi Airport in Bangkok is still closed, so EVA Air changes flight schedule to Bangkok and Europe. EVA Air temporarily take Utapao Airport for the operation for flights to and from and via Bangkok.

Utapao is located approximately 140 kilometres and south from Bangkok. However, facilities at Utapao are very basic and the airport is heavily congested, we apologise in advance for the inconvenience, but ask for your understanding that the airport at Utapao is originally designed for military purpose.

It wasn't just Eva Air that decided to move it's flights, it has, for some time, been Thailand's main airport. From Wikipedia:

With the temporary closure of Suvarnabhumi Airport and Don Mueang Airport in late November 2008 because they had been occupied by anti-government protestors, U-Tapao became for a time Thailand's main supplementary international gateway. Many airlines arranged special flights to and from U-Tapao to ferry international passengers stranded by the closure of the Suvarnabhumi Airport.

As you can imagine, this caused quite a few problems (from the same Wikipedia link):

As many as 100,000 passengers were stranded in Thailand until early December. Although its runway can accommodate large aircraft, U-Tapao's terminals are not designed to handle more than a few flights a day. Travellers were subject to many hardships, and as the security was not up-to-date, some US-bound flights were diverted to Japan and their passengers required to go through a supplementary security check before continuing.

Once upon a time, Penzance (SW England, IATA code PZE, although pretty much just an airstrip) briefly advertised itself as "London Penzance", despite being 306 miles (492km) from London.

I can't find a link, but it was poking fun at the likes of "London Luton" (56km), "London Stansted" (61km), which claim to be alternatives to London City (13km), Heathrow (24km) and Gatwick (45km).

In the United States, the Chicago Rockford International Airport (RFD) is a strong contender. It is located four miles south of Rockford, IL. It is about 90 miles (144 km) from downtown Chicago and about 70 miles (112 km) from ORD. Luckily there is a shuttle from RFD to ORD that runs every 2 hours, which is a trip that takes about 70 minutes.

While mainly a freight airport servicing Chicago, occasionally unlucky passages do end up there thinking they're within shot of the city. I have heard that sometimes the flights to Las Vegas from RFD are cheaper than from MDW or ORD, however have never confirmed this.

Not going to win, but Manchester Boston Regional Airport is about 44 mi (71 km) from the center of Boston (approximate straight-line measurement using Google Maps).

Ryanair's "Oslo" airport (TORP Sandefjord) is 87 km (54 mi) from the center of Oslo.

In general, I'd expect Ryanair's destinations list to include lots of good candidate answers to this question.

Stewart International Airport has been renamed on Feb 21, 2018 to New York Stewart International Airport. It is 72.9 miles from the center of New York City according to Google Maps.

It comes to mind because it's in my hometown.

This example doesn't really count, but it's amusing.

Avelo Airlines recently announced service to the Northern Colorado Regional Airport (FNL), which was known until a few years ago as the "Fort Collins-Loveland Municipal Airport". No problem so far - the airport really is in Northern Colorado, 8 km from the city center of Loveland and 17 km from downtown Fort Collins.

However, their announcement (archived) included the line: "Loveland’s famed skiing is a short 20-minute drive from FNL." Now, the city of Loveland may be very nice, but it's in the plains, not the mountains - it's flat as a pancake and gets only about 120 cm of snow per year, most of which melts within a few days. There's no skiing to be had there, unless maybe you put on your cross-country skis and take a few laps around one of the city parks the morning after a snowfall.

Avelo's PR folks probably confused it with the Loveland Ski Area, located in the mountains near Loveland Pass - which is 114 km from FNL, and would be a "short 20-minute drive" only if you had some sort of rocket sled.

(The only relationship between the Loveland Ski Area and the City of Loveland is that they are both named after William A. H. Loveland).

Not going to beat the other answers, but I wanted to throw in Weeze Airport anyhow, as it was supposed to be named "Düsseldorf-Weeze", but a court blocked the name due to the distance of 60km (Wikipedia claims 83km, the distance on road) from Düsseldorf.

Ryanair still calls it Dusseldorf-Weeze, however - I almost fell for it some time ago.

Another one: Åre Östersund Airport on Frösön, near Östersund.

Initially opened as an air force base in 1926, regular civil aviation commenced there in 1958. It had the name "Frösö flygplats" (Frösö Airport) but was changed in 2006 to "Åre Östersund flygplats" purely as a marketing decision to attract more people to the ski resorts in the Åre area.

By road, it is only 11km from the center of Östersund but over 87km to the center of Åre...

There's an airport in the german city of Memmingen that is often referred to as "Munich West". The issue is that it is just 120km away from Munich.