Another instructor is pushing me out of the classroom right after my class ends I am a graduate student in math in my final year, and for several years have been teaching at my department as a lecturer. This semester, in the same lecture hall there is another lecture that starts 20 minutes after my class ends. It's taught by another instructor from my department. I usually have many students coming to office hours and there are also students asking questions immediately after lecture. Due to other activities, I cannot have office hours right after the lecture this semester and can only stay for about 15 minutes to answer questions.

Many times in the past I had a similar situation and never had any issues with it. This semester the instructor who is teaching right after often arrives 20-15 minutes before her class starts and tells me immediately that I have to go with my students somewhere else. 

I make sure to leave the blackboard clean and take all my stuff away from the instructor's desk before she arrives, but I do believe that I have a right to stay in the classroom after my lecture for at least 5-10 minutes. There is no vacant classroom around, and I don't have time to go with students to my office, which is in a different building.

Last time the instructor told me in front of my students that I don't understand "simple things" and that I am "playing games". When I was talking to one of my students, she stood very close to us and clearly demonstrated that she wanted us out. I tried to explain her that I couldn't go anywhere else due to my time constraints, but she didn't want to listen to me. I really don't understand what "simple" things she meant and what "games" I am playing. 

We leave the board clean. She doesn't need to set up a projector. She can still talk to her students before her class starts, if she wants to (even though it seems like her students don't ask her any questions before their class). So, I don't see how I cause any disruption. 

I had met this woman many times before this semester, but we never talked. I didn't see her talking to other instructors/students much, and she seems to be quite reserved and a bit neurotic. She doesn't want to have any conversation with me regarding the issue.

I felt really offended after last class when she said those things to me in front of my students. What would you do in my case? 

Added later: There are no official rules regarding classroom occupancy between classes. Instructors are supposed to use common sense and be reasonable. For me using 50% of the break time seems reasonable to answer questions after lecture seems reasonable. I agree that for some people it may not.

I don't block the entrance to the classroom. A few students from the next class who come earlier always go ahead and take their seats as soon as my students start leaving the room. I also had one of the students from the next class listening to my explanation to one of those after-class questions and asking me further questions before their class (which is the same class as I am teaching, just a different section). Maybe the instructor got jealous, I don't know.

The entrance to the classroom is from its front (not back), so I do stay in the front. But it is a big lecture hall, and there is a plenty of space in front of the room (the board itself consists of 8 huge panels).

Also, during my career as a grad.student who is also teaching for the department, I have had several observations from experienced professors who are considered to be great teachers at the department and are in charge of undergraduate teaching policy. In my evaluations the fact that there are always several students approaching me with questions after class considered as very positive, meaning that students find me approachable. 

Thank you everyone for answers.
You should first find out what is the standard convention for when you have to vacate the room.  Since the other instructor is in your department, you can take this up at the department level: talk to your advisor and/or a trusted faculty mentor about it.  

In my experience though this is often left a little fungible and people need to be reasonable about it.  In my opinion it is reasonable to expect to have access to the room five minutes before your class begins.  Ideally the previous instructor will have vacated the room at that point; if not, s/he should be occupying the room in a way which doesn't interfere with your own use of it.  (For instance, a lot of times students will not enter a room if an instructor is still writing on the blackboard.  It can be annoying to come to your class a few minutes before you want to start and find everyone waiting out in the hall.)

In the absence of a clear directive about how to split the time, it seems reasonable for the departing instructor to take half of the time, as long as they leave at least five minutes for the other instructor.  In your case 20 minutes is a very nice cushion (in my university it's 15 minutes), to the extent that I find it a little strange that the other instructor is arriving 15-20 minutes early.  What is she doing with that time?  Do more than a few students arrive that early?  If you are using the time to talk to your students and she's using the time for nothing, then I think you are morally in the right and should push back on it a bit.  You should be able to stay for the first half of the 20 minute break.

How should you do that?  As above, I would find out what other people do.  If the standard convention is that you can take those 10 minutes, then talk to the department head (or other departmental authority figure; e.g. whoever schedules the teaching) about it and get their support.  (You don't need to phrase it as a grievance or even identify the other instructor by name.)  Assuming you get backed, then you should pay a visit to the instructor in her office and explain that you looked into the matter, that you have a clear instructional purpose for how to use the time, and that the head (or whoever) has confirmed that this is a standard practice.  Definitely do this outside of the classroom so that you two can have the conversation you need to have not in front of the students and so that there is not a direct confrontation to be resolved.

In my opinion it is probably not worth trying to seek redress for your poor treatment by this instructor.  We are not obligated to be nice to each other; it's just better if we are.  If you respond to the treatment in a professional way and show that you are neither going to get pushed around nor retaliate in any personal or emotional way, then you're basically just rising above.  What some random person who you're only going to see for one more semester thinks of you isn't really so important, is it?  Especially if you're confident that you're in the right?