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Related: If I cast Banishment on myself while in a demiplane, where exactly do I exit?
If I am trapped in the extradimensional space of a bag of holding, can I escape by casting Banishment on myself? Is the space inside of a bag of holding considered to be a plane with respect to this spell, or is it more a place that is not part of any plane?
Demiplanes are discussed in the DMG in the chapter on planes, under the “Other Planes” heading — a demiplane is absolutely a plane in its own right. And “extradimensional space” and “demiplane” mean the same thing, per that section (DMG, p. 68):
Other Planes
[…]
Demiplanes
Demiplanes are extradimensional spaces that come into being by a variety of means and boast their own physical laws.
(For more discussion of the nature of the things called demiplanes/extradimensional spaces/pocket dimensions, see this question on that exact subject.)
Basically, everywhere in the D&D cosmos is a plane of some kind or part of a plane. Extradimensional spaces must exist somewhere, are explicitly not part of other planes (ibid.); therefore all extradimensional spaces are demiplanes (by the process of elimination), and planes in their own right. Magic items that connect to demiplanes are better thought of as keys or gates to another plane.
And we know that if you cast Banishment in a demiplane, the target goes back to its home plane (to an unpredictable location), and you can cast Banishment on yourself.
So unless you're a native to that bag's demiplane, yes, casting Banishment will let you escape. However, be prepared to show up anywhere on your native plane. If you native plane is very large, you might end up escaping into an even more difficult predicament.
It doesn't matter if it is a demiplane or not. Spells do what they say. The spell banishment doesn't say you have to be on a plane or demiplane for it work, it just says:
If the target is native to a different plane of existence that the one you’re on...
The Handy Haversack description makes it clear that the inside of a bag of holding is extradimensional space:
Placing the haversack inside an extradimensional space created by a bag of holding...
The inside is clearly in extradimensional space. Extradimensional means outside of dimensions, that is, it is not on the prime material plane. If you are in extradimensional space, you not on the prime material plane. When banished, you'll return to the prime material, if that is the plane to which you are native.
... the target is banished with a faint popping noise, returning to its home plane.
which would mean anywhere the DM wants to place the target in the prime material plane.
The DMG stats the definition of Demiplane as:
Demiplanes are extradimensional spaces that come into being by a variety of means and boast their own physical laws.
We know from above it meets the first criteria, it is extradimensional space.
The second criteria is that it has its own physical laws that are stated in the description.
If the bag is overloaded, pierced, or torn... If the bag is turned inside out... Breathing creatures inside... (etc.)
Being an extradimensional space with its own physical laws makes it a demiplane.
It goes on to say:
Theoretically, a plane shift spell can carry travelers to a demiplane, but the proper frequency required for the tuning fork would be extremely hard to acquire. The gate spell is more reliable, assuming the caster knows of the demiplane.
Which just confirms the banishment statement from above.
Provided you can squeeze into the Bag, with its 2ft diameter opening; if you were inside a bag of holding, or any other extra-dimensional space, the relevant part of Banishment is as follows:
If the target is native to the plane of existence you’re on, you banish the target to a harmless demiplane. ... If the target is native to a different plane of existence than the one you’re on, the target is banished with a faint popping noise, returning to its home plane.
Extra-dimensional Space is not necessarily a plane in it's own right, the DMG contains only a small paragraph on Demiplanes, most notably:
Demiplanes are extradimensional spaces...
However nothing is said about whether all extra-dimensional space is on, or is entirely, a plane/demiplane. "All Dogs are Mammals" does not mean all "Mammals are Dogs".
So there are a few options:
It's down to your DM to decide which of these options is taken, bearing in mind that if, as per the above rulings, you count as being native to the plane you're on when you cast it, you would be sent to a harmless demiplane and become incapacitated as per the description of Banishment, thus immediately losing concentration on your spell, and returning to the inside of the Bag.
A spell that would definitely work? Plane Shift:
You and up to eight willing creatures are transported to a different plane of existence.
This has no prerequisites regarding your current plane, and requires no concentration/Saving Throw, so you would be free to escape the bag.
*This answer has been edited to take into account the comments from multiple users.