(Before I start I would like to clarify that I am not a PUA and do not really want to be one, but I find the PUA articles and this forum pretty useful.)

So, I had a pretty good first date with a really nice girl (HB9) a bit more than a week ago. She was totally attracted to me, mirroring me and smiling all the time.
I was funny, cocky and interesting, but tried to be natural and not playing Jedi games with her, however there are two issues that I do not know how to handle:

A. Trust
She has a very big issue with trusting people. Maybe because she has some kind of bad memory or experience.
- When I first met her, she did not want to kiss me. She just simple shoke hand with me. (at the end she gave me a kiss)
- She hardly wanted to walk with me in a street which was a bit darker than a usual one.
- She does not like big parties/events with lot of people. She prefers house parties with her friends.
- She did not want to give me her phone number before the first date. She called me once, but she hide her phone number.

B. Time
I cannot put it a better way, but she is workaholic. I think If i could get close to her she would be willing to make more free time.
Beside her daily job she is also a private teacher. Usually she has 2 free nights per week as I see.

...

At the end of the first date, she tried to setup a second date with me and also give me her phone number, but I told her not to give it to me immediately.
I was trying to look or position myself as the "Prize" and not a needy or easy guy, so a bit different.

Some days after the first date I tried to arrange another date with her, but she is always too busy. For example today She is going to a birthday party.
I am not expecting from her to invite me, because I know I should gain her trust first.

I am also going to a party tonight and I was thinking about sending her a funny sms or call her to come to dance with me.
She really likes to dance, so this would totally make her feel excited, but I also do not want to make her choose between her party and mine, because I know I would loose.

My point is that I do not want to look needy by calling her or sending sms to her all the time.

What should I do to gain her trust and/or make her chase me?

You're a PUA, get over it

I think it's the same problem I've seen and experienced many times. You had an awesome time together, but instead of acknowledging it and expressing an interest in seeing her again (in an alpha way of course), you play it "too cool", and go the opposite way.

You should have taken her number, and then made a flirty joke about when you'll get back to her.

Now she just thinks you're a player. Only way out is to meet again randomly or through something else, and start building the attraction all over again.