Is it permitted to purposely extend Shabbos to Sunday night when Tisha B'av is observed on that Sunday?
We see that regarding Jews traveling near the international dateline, there are Poskim who said to keep 2 days of Shabbos.
I saw in this week's Parsha Sheet (Shabbos 9 Av 5779) that Rav Moshe Sternbuch's Talmidim print, that he paskens that one may not extends Shabbos for more than a few minutes after one's customary "Shabbos-ending time" - and at the very latest one has to end Shabbos 3 - 4 minutes after the Rabbeinu Tam time.
He did not quote any sources.
Extending Shabbat up to Sunday night would create a conflict between two halachot.
On one side, we need to fast on Tisha b'Av
It is a positive commandment, ordained by the Prophets, to fast on those days on which tragic events occurred to our forefathers. The purpose of the fast is to stir our hearts to reflect on the ways of repentance, and to serve as a reminder of our own evil deeds, and the deeds of our forefathers, which were as reprehensible as ours, and caused all these troubles for them as well as for us. By remembering these events we will improve our ways [...] (Kitzur Shulchan Aruch 121:1)
On the other side we cannot fast, grieve or mourn on Shabbat
It is forbidden to fast on Shabbos for the sake of fasting, even for a short time [..] It is forbidden to grieve, heaven forbid, over any misfortune, may the Merciful One save us, but you should seek mercy from the Merciful One (Kitzur Shulchan Aruch 77:20-21)
As such Shabbat and Tisha b'Av are fundamentally incompatible.
One commenter asked how to reconcile the above with the idea one can extend Shabbat by a few minutes/hours of tosefet Shabbat even on the night of Tisha b'Av. I believe there is a significant difference between not eating for a few hours of tosefet Shabbat in the evening, when not yet very hungry, which is not called fasting, and not eating all of Sunday which is called fasting. I do not know where the exact boundary is but since the question is until Sunday night, I believe it is clear that not eating all of Sunday is fasting. And since fasting is prohibited on Shabbat, extending Shabbat to Sunday night is in conflict with the fasting of Tisha b'Av.
(regarding Yom Kippur on Shabbat, there is an explicit pasuk for this).
There's nothing stopping you from keeping Shabbos for as long as you like(*), however, you can't eat, drink or wash after sunset.
This is documented by the Remo in תקנ"ב and by the Kitzur Shulchan Aruch inסימן קכה - דיני תשעה באב שחל בשבת או באחד בשבת where he says:
וְצָרִיךְ לְהַפְסִיק מִבְּעוֹד יוֹם, כִּי בְּבֵין הַשְּׁמָשׁוֹת אָסוּר בַּאֲכִילָה וּשְׁתִיָּה וּרְחִיצָה
It seems that as long as you are keeping Shabbos you don't have to remove your shoes, since he continues:
אַךְ הַמִּנְעָלִים לֹא יַחֲלוֹץ עַד לְאַחַר בָּרְכוּ
He doesn't discuss when you have to start sitting low/on the floor.
All the above is applicable until you begin Maariv - at which point it's Tisha B'av for you. If you're a Chazzen then after בָּרְכוּ, else, when you are about to say וְהוּא רַחוּם, as the Kitzur taught us:
אַךְ הַמִּנְעָלִים לֹא יַחֲלוֹץ עַד לְאַחַר בָּרְכוּ. וּשְּׁלִיחַ הַצִבּוּר חוֹלֵץ קֹדֶם שֶׁמַּתְחִיל וְהוּא רַחוּם, כְּדֵי שֶׁלֹּא לְבַלְבֵּל דַּעְתּוֹ
Halocho does not seem to discuss what somebody does if he didn't daven Maariv and wants to keep Shabbos instead. That's not unusual that Halocho ignores those who ignore Halocho.
And don't forget to say בָּרוּךְ הַמַּבְדִּיל בֵּין קֹדֶשׁ לְחֹל when you do decide to end Shabbos and start week. Actually, the Kitzur recommends one says the longer version:
בָּרוּךְ הַמַּבְדִּיל בֵּין קֹדֶשׁ לְחֹל, בֵּין אוֹר לְחֹשֶׁךְ, בֵּין יִשְֹרָאֵל לָעַמִּים, בֵּין יוֹם הַשְּׁבִיעִי לְשֵׁשֶׁת יְמֵי הַמַּעֲשֶׂה, בָּרוּךְ הַמַּבְדִּיל בֵּין קֹדֶשׁ לְחֹל
Also: You should not learn anything not-Tisha B'av related at this point, since even during the afternoon you were limited to what you could learn.
תִּשְׁעָה בְּאָב שֶׁחָל בַּשַׁבָּת יֵשׁ לְהַחְמִיר שֶׁלֹּא לִלְמֹד כִּי אִם דְבָרִים הַמֻּתָּרִים לִלְמֹד בְּתִשְׁעָה בְּאָב, וְלָכֵן אֵין אוֹמְרִים פִּרְקֵי אָבוֹת. אֲבָל לִקְרוֹת הַסִּדְרָה שְׁנַיִם מִקְרָא וְאֶחָד תַּרְגּוּם, מֻתָּר וּמִכָּל שֶׁכֵּן קֹדֶם חֲצוֹת הַיּוֹם. וְאִם חָל (בְּ) עֶרֶב תִּשְׁעָה בְּאָב [בַּשַׁבָּת] אָסוּר לְאַחַר חֲצוֹת הַיּוֹם בְּלִמּוּד כְּמוֹ בִּשְׁאָר תִּשְׁעָה בְּאָב.