Q: If a Muslim is incarcerated and ends up in a small jail cell that has a bunk bed right next to a toilet and sink, and is locked down for the majority of the day (only 1-3 hours of free time per day on a rotating schedule), how can he pray Salah in an unclean place when he is locked down during the prescribed time for prayer? Will he have to make up these missed prayers when he does get let out and have an opportunity to pray in a clean place? Or are there exceptions to the rule? What becomes of missed prayers of inmates who can't pray in a clean place due to being locked down for weeks or months at a time?
A: If the inmate can delay their prayer, without exceeding its chosen time, in order to pray in a clean location outside of the cell, they must do so.
If the inmate is not permitted to leave the cell for the duration of one or more prayers and the cell floor has some area that is clean, they must pray in that location.
If it is completely soiled with legal impurity (not merely dirt) which cannot be purified or covered with something clean and thick, they are to prostrate on any dried impurity. If it is all moist they are to gesture in lieu of prostrating to the best of their ability while sitting on their feet so that nothing else comes in contact with the impurity.
Prayer in this condition must be made on time and is not required to be made up.
In this case, voluntary prayers are not permitted.
Source: Sharh al-Muntaha, Hidayat al-Raghib