Learn Hardcore German: Ich bleibe auf der Bank sitzen.

auf is a so‑called two-way preposition (Wechselpräposition). With these, the case depends on whether there is movement into a new place or just location:

Accusative = movement to a destination (answering “where to?”)
Dative = position / location (answering “where?”)

In your sentence, you are already on the bench and staying there, so it’s a location:

Wo? (Where?) – auf der Bank → dative

If there were movement onto the bench, you would see accusative (usually with another verb):

Ich setze mich auf die Bank. – I sit down on the bench. (movement to the bench)

So, with bleiben (to remain), you normally get dative:
Ich bleibe auf der Bank sitzen. – stay seated on the bench.

Why is it der Bank and not die Bank?

The noun Bank is feminine:

Nominative: die Bank
Accusative: die Bank
Dative: der Bank
Genitive: der Bank

Because auf here takes the dative (it’s a stationary location), you must use the dative feminine form:

auf der Bank (not auf die Bank)

So you’re seeing a case change, not a change in gender. The gender is still feminine; only the article form changes in the dative.

Does Bank here mean “bench” or “financial bank”? How do I know?

In this sentence, Bank clearly means bench. You can tell from:

The preposition
- For sitting on a bench in a park, Germans say auf der Bank sitzen.
- For being inside a financial institution, Germans say in der Bank sein (in the bank building).
The verb and context
- You normally sit on a bench: auf der Bank sitzen.
- You would not really say in normal conversation: “Ich bleibe auf der Bank sitzen” about a financial bank; that would sound odd.

So:

die Bank = the bank (institution) → often used with in (in der Bank).
die Bank = the bench → often used with auf (auf der Bank).

Same spelling, different meanings; the preposition + context show which one is intended.

Why does sitzen go at the very end of the sentence?

In a simple main clause, German normally follows this pattern:

Some element in position 1 (often the subject, but not always)
The conjugated verb in position 2
Other sentence parts
Any infinitive(s) or participle(s) at the end

In your sentence:

Ich – subject in position 1
bleibe – conjugated verb in position 2
auf der Bank – prepositional phrase in the “middle”
sitzen – infinitive at the end

So: Ich | bleibe | auf der Bank | sitzen.

This “frame” of bleibe … sitzen around other material is very typical of German main‑clause word order, often called the Satzklammer (sentence bracket).

Could I say Ich bleibe sitzen auf der Bank instead?

You can put it that way, and it’s grammatically possible, but:

Ich bleibe auf der Bank sitzen. is the most natural and neutral-sounding order.
Ich bleibe sitzen auf der Bank. sounds a bit more marked and could put a slight emphasis on “on the bench” as a kind of afterthought or clarification.

In practice, everyday spoken German strongly prefers:

Ich bleibe auf der Bank sitzen.

Use the “bleibe … sitzen” frame with the location phrase in between. That’s what native speakers will expect.

What’s the difference between Ich sitze auf der Bank and Ich bleibe auf der Bank sitzen?

Both describe you on the bench, but the focus is different:

Ich sitze auf der Bank.
- Simple description of your current position.
- Roughly: “I am sitting on the bench.”
- No special focus on what happens next.
Ich bleibe auf der Bank sitzen.
- Emphasises that you do not get up / you stay there.
- Roughly: “I remain sitting on the bench / I stay seated on the bench.”
- Often implies some contrast: you could get up, but you decide not to (or are told not to).

So bleibe … sitzen expresses continuation of the state, not just the state itself.

Is sitzen bleiben / sitzenbleiben also the verb that means “to repeat a school year”? How is that related?

Yes. sitzen bleiben (often also written sitzenbleiben) can mean:

Literal: remain seated
- Ich bleibe auf der Bank sitzen. – I stay seated on the bench.
Figurative / school context: have to repeat a year
- Er ist in der achten Klasse sitzen geblieben / sitzengeblieben.
  – He had to repeat 8th grade.

Same combination of verbs, but in (2) it has developed a fixed, idiomatic meaning.

In your sentence, context makes it obviously literal (you’re physically on a bench), so it just means “remain seated.”

Can I omit auf der Bank and just say Ich bleibe sitzen?

Yes.

Ich bleibe sitzen. = I stay seated / I remain sitting (where I am).

This works whenever the place is already clear from the context, or simply doesn’t matter.
Adding auf der Bank just specifies where you’re staying seated.

How would I say “I stay sitting on it” instead of repeating Bank?

You can replace auf der Bank with a da‑compound:

Ich bleibe darauf sitzen. – I stay sitting on it.

Here:

auf der Bank → darauf (on it)
Word order stays: Ich bleibe darauf sitzen.

If the object were mentioned earlier (for example: die Bank, den Stuhl, die Kiste), darauf would naturally refer back to that without naming it again.