I assume that your are a Malaysian national and resident, and that you have not lived for any period of time in the UK on a long terms visa status (eg. student/workers/family visa).
In order to claim asylum in this country one must be unable to go back to your home because of a well founded fear of persecution because of your race, religion, nationality, political opinion or membership of a social group.
You would have to argue that the Malaysian attitude to your membership of a social group (ie; an unmarried mother) is such that you would suffer persecution. The difficulty is that Malaysia is not generally seen as a County with the type of regressive attitude that you suggest so you would face an uphill battle to convince the UKBA of the validity of your right to claim asylum by producing evidence (newspaper reports/articles/statements) to prove your case. It will not be easy.
I attach a link the UKBA's page on how to claim asylum:-
The alternative would be to attempt to make an discretionary application to the UKBA to exercise their discretion to permit you to stay on the basis of your dependency of the child. You would need a solicitors for this however as it is quite complicated.
You could make an application to Court for an order that you be permitted to return to Malaysia with the child, but again this does not get around your fears of returning home.
I would focus first on obtaining documentary evidence based on your fears and collecting them together. Take this to an immigration solicitors and ask for a fixed fee consultation to get specific advice on the merits of your asylum claim. You can then make a decision based upon the likelihood of success as explained to you based on your evidence.
It sounds an awful situation, I am very sorry for you.
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