How should people fast whose day is very long and the sun never sets for them?

Those whose night and day add up to twenty-four hours should
fast that day, whether it is short or long, and that is acceptable from
them, praise be to Allaah, even if the day is short. But those for whom the
night or day is longer than that, such as six months, should work out the
time for fasting and prayer, as the Prophet enjoined should be done on the
day of the Dajjaal which will be like a year, and his day which will be like
a month, or like a week. The times for prayer should be worked out. The
Council of Senior Scholars in the Kingdom examined this matter and issued a
statement no. 61, dated 12/4/1398 AH which says: 

Praise be to Allaah, and blessings and peace be upon His
Messenger and his family and companions. 

Firstly: Whoever lives in a land where the people can
distinguish night from day by the rising of the dawn and the setting of the
sun, but their day is very long in summer and very short in winter, are
obliged to offer the five daily prayers at the times that are known in
sharee’ah, because of the general meaning of the verses in which Allaah says
(interpretation of the meaning): 

“Perform As‑Salaah (Iqaamat‑as‑Salaah)
from mid‑day till the darkness of the night (i.e. the Zuhr, ‘Asr, Maghrib,
and ‘‘Isha’ prayers), and recite the Qur’aan in the early dawn (i.e. the —
Fajr morning prayer). Verily, the recitation of the Qur’aan in the early
dawn (i.e. the morning — Fajr prayer) is ever witnessed (attended by the
angels in charge of mankind of the day and the night)”

[al-Isra’ 17:78] 

“Verily, As‑Salaah (the prayer) is enjoined on the
believers at fixed hours”

[al-Nisa’ 4:103] 

And it is proven from ‘Abd-Allaah ibn ‘Amr ibn al-‘Aas (may
Allaah be pleased with him) that the Messenger of Allaah (peace and
blessings of Allaah be upon him) said: “The time for Zuhr is when the sun
passes its zenith and the shadow of a man is equal in length to him, so long
as ‘Asr has not come. The time of ‘Asr is so long as the sun has not turned
yellow. The time for Maghrib is so long as the twilight has not yet
disappeared. The time for ‘Isha’ is until halfway through the night. The
time for Fajr is from dawn, so long as the sun has not yet risen. When the
sun rises, then refrain from praying, for it rises between the horns of the
Shaytaan.” 

And there are other ahaadeeth which define in word and deed
the times for the five daily prayers, but no differentiation is made between
long days and short days, or long nights and short nights, so long as the
times of prayer can be determined by the signs explained by the Messenger of
Allaah (peace and blessings of Allaah be upon him). 

This has to do with the definition of the times of prayer. 

With regard to the times for fasting the month of Ramadaan,
those who are accountable must refrain from eating, drinking and everything
else that breaks the fast on every day of the month from dawn till sunset in
their country, so long as the day can be distinguished from the night in
their country, and the total of day and night is twenty-four hours, and it
is permissible for them to eat, drink, have intercourse and so on only
during their night, even if it is short. The sharee’ah of Islam is universal
and applies to all people in all lands. Allaah says (interpretation of the
meaning): 

“and eat and drink until the white thread (light) of dawn
appears to you distinct from the black thread (darkness of night), then
complete your Sawm (fast) till the nightfall”

[al-Baqarah 2:187] 

If a person is unable to fast the whole day because it is too
long, or because he knows from signs or by experience or on the advice of a
trustworthy, skilled doctor, or he thinks it most likely that fasting will
cause him to die or to become severely ill, or will make his sickness worse
or will slow down his recovery from sickness, then he should break his fast,
and make up the days that he did not fast during any month when he can make
them up.  Allaah says (interpretation of the meaning): 

“So whoever of you sights (the crescent on the first night
of) the month (of Ramadan i.e. is present at his home), he must observe Sawm
(fasts) that month, and whoever is ill or on a journey, the same number [of
days which one did not observe Sawm (fasts) must be made up] from other
days”

[al-Baqarah 2:185] 

“Allaah burdens not a person beyond his scope”

[al-Baqarah 2:286] 

“and has not laid upon you in religion any hardship”

[al-Hajj 22:78] 

Secondly: 

The one who lives in a land where the sun does not set during
the summer and the sun does not rise during the winter, or he lives in a
land where the day lasts for sixth months and the night lasts for six months
for example, should offer the five daily prayers during each twenty-four
hour period, and he should try to work out their times, based on the closest
land to him where the times of the five prayers are distinct from one
another, because of what it says in the hadeeth about the Isra’ and Mi’raaj,
that Allaah enjoined upon this ummah fifty prayers every day and night, then
the Prophet (peace and blessings of Allaah be upon him) kept asking his
Lord to reduce it until He said: “O Muhammad, they are five prayers each day
and night.” Narrated by Muslim (162). 

And it was narrated that Talhah ibn ‘Ubayd-Allaah (may Allaah
be pleased with him) said: A man from among the people of Najd, with
disheveled hair, came to the Messenger of Allaah (peace and blessings
of Allaah be upon him), and we could hear the sound of his voice but we
could not understand what he was saying, until he drew close to the
Messenger of Allaah (peace and blessings of Allaah be upon him), and he
was asking about Islam. The Messenger of Allaah (peace and blessings of
Allaah be upon him) said: “Five prayers each day and night.” He said, Do I
have to do any (prayers) other than that? He said, “No, unless you do them
voluntarily

Narrated by al-Bukhaari (46) and Muslim (11). 

And it is proven that the Prophet (peace and blessings
of Allaah be upon him) told his companions about the Dajjaal, and they said:
How long will he stay on earth? He said: “Forty days, a day like a year, a
day like a month, a day like a week, and the rest of the days like your
days.” We said: O Messenger of Allaah, on that day which is like a year,
will the prayers of one day be sufficient for us? He said: “No. Work out the
time (for prayer).”  Narrated by Muslim (2937). He did not regard that day
that will be like a year as being a single day in which five prayers would
be sufficient, rather he enjoined praying five prayers every twenty-four
hours, and he commanded them to space them out during the time based on the
times on an ordinary day in their country. So the Muslims in the country
asked about here have to define the times for prayer based on the closest
country to them in which night is distinct from day, and thus the times for
the five daily prayers may be known by their shar’i signs (in the closest
country) in every twenty-four hour period.  

Similarly they have to fast the month of Ramadaan, and they
should work out the beginning and end of the month of Ramadaan, and the
times to start and end the fast every day, and the time of dawn and sunset
every day, in the closest country to them where night is distinct from day
and the total of night and day is twenty-four hours, because of the hadeeth
of the Prophet (peace and blessings of Allaah be upon him) about the
Dajjaal that we have quoted above, in which he told his companions how to
work out the times of prayer. There is no difference in this case between
fasting and prayer. And Allaah is the Source of strength. May Allaah send
blessings and peace upon our Prophet Muhammad and his family and
companions. 

Council of Senior Scholars. End quote. 

Majmoo’ Fataawa al-Shaykh Ibn Baaz
(15/292/300).