The Council of Senior Scholars in the Kingdom of Saudi Arabia issued a statement on the determining of the times of prayer, determining the start of dawn each day, and the end of the dawn in Ramadan, in countries in which the sun does not set during the summer and does not rise during the winter. The full statement could be referred to in the link below. The relevant part of the statement is in the following:
the Muslims in the country mentioned in the question have to define the times of prayer in that land, basing those times on the closest country in which the night and day can be distinguished from one another and the times of the five daily prayers are known according to the signs described in sharee'ah (Islamic law), within each twenty four hour period.
Similarly they also have to fast Ramadan. They can set the time for their fast and determine the beginning and end of Ramadan and the times of starting and breaking the fast each day by the dawn and sunset each day in the closest country in which night can be distinguished from day. The total period must add up to twenty four hours, because of the hadeeth of the Prophet (peace and blessings of Allah be upon him) about the Dajjaal mentioned above, in which he told his companions how to determine the times of the five daily prayers. There is no difference in this regard between fasting and prayer.
Whoever resides in a country where the night can be distinguished from the day by the onset of dawn and the setting of the sun, but the day is extremely long in the summer and extremely short in the winter, is obliged to perform the five daily prayers at the times known in sharee'ah. That is because of the general meaning of the words of Allah (interpretation of the meaning):
"Perform As‑Salaat (Iqamaat‑as‑Salaat) from mid‑day till the darkness of the night (i.e. the Zuhr, 'Asr, Maghrib, and 'Ishaa' prayers), and recite the Qur'aan in the early dawn (i.e. the - Fajr morning prayer). Verily, the recitation of the Qur'an 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‑Salah (the prayer) is enjoined on the believers at fixed hours"
[al-Nisa' 4:103]
See the link below for full statement with more details.