What if one begins their shmoneh esrai just before the latest time for davening, but that time comes while they are in the middle of shmoneh esrai. Does he continue to the end even after the zman passes, or does he stop? This goes for shacharit (both z’man t’fila and chatzot), mincha (shkia), and maariv (netz).
For all three tefillos he should finish before the time is up, however if he didn’t he should continue and finish davening.
For Shacharis- See sources that he should continue.
For mincha- even though it is almost shkia, he should daven even though part of it will be after shkia, because bedieved there are poskim that say that the time extends further.
For maariv he may continue, since there are opinions that one can daven after alos hashachar, although we don’t daven then unless one was it was an accident, however for this situation he can rely on it to continue davening.
Shacharis- Ehsel Avrohom Butchatch (Tinyana) 89-1, Shut Eretz Tzvi 121, Shearim MItzuyanim B’halacha18 ftnt 2.
Mincha- M:B 234-14, E:A Butchech (tinyana) 234, Aruch Hashulchan 101-5. However see Orchos Rabeinu 3 pg. 225 from the Chazon Ish that if he thinks he will miss shkia he shouldn’t start. (However this might be specifically in Eretz Yisroel that the sun sets quickly there, see Eishei Yisroel chap 27 ftnt 14.)
Maariv- Piskei Teshuvos 235-12