Yes, death is predetermined.
Substantiating with two Pauranika sources -
1. The Garuda Purana (abridged)
Death And After
Continuing with his narration, Lord Krishna said–” O Garuda! Death
comes at the predetermined time, which remains fixed and unaltered
under all circumstances. A man’s sense organs become weak and his body
feeble, which gets inflicted with numerous diseases in his old age. At
the time of his death, man experiences unbearable pain and he begins
to loose his consciousness. Yamdoots arrive and begin to retrieve the
soul from the body and this aggravates the pain. Ultimately, the soul,
which is not more than the size of a thumb, reluctantly comes out from
the body as the attachment with the world exists even after his death.
But, a virtuous person does not go through all these painful
experiences at the time of his death. The soul has to take birth in
different species as well as go through cycles of countless births,
deaths and rebirths in order to taste the fruits of his Karmas.
2. Padma Purana:
Chapter 81 - Destiny is Irresistible
to 40. ..... Destiny brings about humiliation by remaining wheresoever (it pleases). It makes a man a donor or a suitor.
Destiny holds everything—all immobile and other beings (living) in heaven or on the earth. Destiny alone is this world. It is without
origin and death and is the greatest cause of the world. Destiny
ripens the worlds as the fruit laid on a tree. Hymns, penance,
charity, friends or relatives are not able to protect a man oppressed
by destiny. It is not possible to overcome the three nooses of
destiny: marriage, birth and death—when and where one would have
these, and with whom or through whom. As the clouds in the sky are
moved by the wind, so the world is moved by destiny united with (the
fruits of) the deeds (of beings). ....
to 67. But the destiny, which, united with Karman (the deeds), is adored by men, would (only)' urge Karman (the fruits of the deeds),
and does not create it. In the human (world), calamities, misfortunes,
serpents and diseases, move (in accordance with) their being decided
by (one’s) deeds. All those that are the causes and means of
happiness, being mixed with merit, are united with (the fruits of)
deeds. They would not see (i.e. would not care for what is) auspicious
and (what is) inauspicious. (Obscure!) relatives united with (the
fruits of) deeds may exchange them; but (the fruits of) deeds (alone)
urge men on to happiness and unhappiness in this world. As gold or
silver has its nature fixed, similarly a being is bound in accordance
with his deeds. These five are produced (i.e. decided) when a man is
just in (his mother’s) womb: his life (i.e. longevity), deeds, wealth,
learning and death. As an agent (potter) fashions from a lifeless
lump whatever he wants to fashion, in the same way deeds performed
before follow the doer. One becomes a god, or a man, or a beast, or a
bird, or a lower animal, or an immobile object, according to one’s
deeds. He always enjoys in accordance with that only which is
accomplished by himself—unhappiness results from one’s own deeds;
happiness results from one’s own deeds. Obtaining the bed of womb, he
enjoys the fruit of his deeds of the previous body (i.e. done in the
previous existence). On the earth men never (i.e. can never) give up
the fruits of their deeds.
So, to the question -
"Is death always part of prarabdh (i.e. fruit of actions of previous birth). Or Can death be caused by the fruit of actions of this birth also ?"
Yes, death is a part of our destiny, pre-determined mostly based on our own actions, generally done in our previous lives, as discussed in detail in the above two excerpts.