She fasted for six days of Shawwaal and wants to carry on fasting

Firstly:

There is
nothing wrong with a person joining a voluntary fast to a made-up fast or six
days of Shawwaal, because of the general meaning of the evidence which encourages
offering voluntary fasts without a separation between the voluntary fast and
the made-up fast.

Secondly:

If what is
meant by your continuing to fast after completing the six days of Shawwaal is
observing voluntary fasts continually until the end of Shawwaal or for certain
days that you want to fast voluntarily, seeking thereby the pleasure of Allah,
there is nothing wrong with that, so long as neither of you will be harmed by
that and neither of you will fail to fulfil the rights of the other.

However the
best teaching concerning that is not to fast for an entire month except the
month of Ramadan; rather the days when one fasts should be mixed with days when
one does not fast, and vice versa, as was the practice of the Prophet
(blessings and peace of Allah be upon him).

It was
narrated that ‘Aa’ishah (may Allah be pleased with her) said: The
Messenger of Allah (blessings and peace of Allah be upon him) used to fast
until we said that he would not break the fast, and he used to refrain from
fasting until we said that he would not fast. And I never saw the Messenger of
Allah (blessings and peace of Allah be upon him) fast any month in full except
Ramadan, and I never saw him fast more than he did in Sha‘baan.

Narrated by
al-Bukhaari, 1969

Shaykh Ibn
Jibreen (may Allah have mercy on him) said: It is permissible to fast
continually on consecutive days, then not to fast continually on other
consecutive days. The evidence for that is the hadeeth quoted in the question,
because that is voluntary and mustahabb.

End quote from
Fataawa ash-Shaykh Ibn Jibreen

But if what is
meant is continuing to fast until the next year apart from the two days of Eid
and the days of at-tashreeq (the three days following Eid al-Adha), this is
what the scholars call lifetime fasting (sawm ad-dahr) and the ruling on it is
that it is makrooh according to the correct scholarly opinion.

For more
information, please see the answer to question no. 144592

And Allah knows best.