May Allah bless all of us with right doings!
My question is that: Is a man (who is breastfed) allowed to marry the younger sister of his foster sister? (breastfed only once). Please quote references from the Qur'an or hadith.
Is it permissible for him to marry the younger sister of his foster sister?
No. The prohibition is clearly mentioned in the following verse:
[4:23] Prohibited to you [for marriage] are your mothers, your daughters, your sisters, your father's sisters, your mother's sisters, your brother's daughters, your sister's daughters, your [milk] mothers who nursed you, your sisters through nursing, your wives' mothers, and your step-daughters under your guardianship [born] of your wives unto whom you have gone in. But if you have not gone in unto them, there is no sin upon you.
And [also prohibited are] the wives of your sons who are from your [own] loins, and that you take [in marriage] two sisters simultaneously, except for what has already occurred. Indeed, Allah is ever Forgiving and Merciful.
However, in this case, the man mentioned in the question is breastfed only once. In such a case, it is permissible for him to marry her because she is not considered to be his foster sister. That is because, it takes at least five feedings to make someone to be considered as breastfed. Shaykh Muhammad ibn Saalih al-‘Uthaymeen said:
...the breastfeeding which has an effect (of creating the relationship of mahram) is that which consists of five feedings or more within the first two years, before weaning. [Fataawa Islamiyah, 3/338]
This view is also accepted by the scholars of Islamweb. Islamweb states in fatawa 264067:
The scholarly opinion adopted by Islamweb is that in order for the breastfeeding to be a basis for the forbidding of marriage, the number of breastfeeding sessions should be five or more, as highlighted in Fatwa 83788.
And Allaah knows best.
Whom will be a mahram by breast-feeding
Well among those who are prohibited for marriage quoted in surat an-Nisa' (4:23) you'll find:
... your [milk] mothers who nursed you, your sisters through nursing...
Assuming the breast-feeding is counted mas established: Then from that point of View scholars would tell you that the mothers of the foster-Mother (whom are considered as her roots), her daughters (who are considered as her branches) would be mahram for him (You can read about it here in Arabic and read for example in Sahih Muslim 1 & 2), while her (the foster-mother's) sisters or aunts wouldn't be mahram (as they are only her surrounds).
This means that the sister of the foster sister no matter if she was younger or older is haram for marriage.
When one would be considered as breast-fed
There's a Hadith in Sahih Muslim twice one with Imam Malik in the narrator Chain and once again by a Student (Al Qa'nabi) of Imam Malik, but on the authority of Sulayman ibn Bilal.
You could read the details in my answer about the interpretation of this Hadith. There I quoted that there one can find that most scholars say that this Hadith is an ahaad and it couldn't be taken as an evidence for naskh in the Qur'an so most (earlier) scholars would say that even one suckling is enough to be counted as breast-feeding, while Shafi'i's (and those who followed them) took the Hadith as is and said at least a couple of suckling if not 5 would be counted as breast-feeding.
But I'm not a scholar and only quoting what I found, so you should ask somebody who knows it well!
See also this answer which shows the definition of one suckling according both opinions.