Is Zakah payable on the Mahr an woman get during her marriage? What if the Mahr is not fully paid? Is it an obligation to pay zakah on the portion of mahr that has been paid?
As said in my comment above zakat is payable for an amount that reaches the nisab (a fixed amount of equal to a given amount of about 585 grams of silver or about 85 grams of gold, due to the fact that the prices of silver and gold have changed from the time of the prophet () until now scholars have different opinions on what to take as a guidline some say gold while others say the prescribed amount of silver is fine) and was kept for one hawl (1 lunar year).
If the amount given for mahr also reaches this value one must pay zakat. If the mahr was not yet (fully) paid its is considered as debt on your husband. So it is basically considered as your money lend to your husband.
Let's assume you've got married on the 1st of Muharram and got in possession of your mahr or a part that reaches the nisab and have not spent from it during the following year then you must pay 2.5 % of it as a zakat on the next 1st of Muharram. In case the money was not yet given by your husband and there's no fear that this would affect your marriage (fear he would divorce you or get angry if you insisted on having it paid or given) etc. you may ask him to pay the zakat of your mahr.
So if you've got only one part of it and this part doesn't reach the nisab while the full mahr would reach it again you either may pay zakat (based on the full mahr -note there's a condition so read until the end of the post-) once after the lunar year has passed or ask your husband to pay it. If you couldn't convince your husband to pay the zakat of your mahr you wait until you have it in your hand and pay the zakat only once for the past years (its is not necessary to calcualte the "missed" zakat for all the passed years you pay only one zakat based on 2.5% of the full amount of the mahr).
Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) was asked about the dowry of a woman who had been married for many years but she was not able to ask her husband for it lest they become separated, then she was compensated for her dowry with some real estate or by being given the dowry after many years. Does she have to pay zakaah for the past years or when one year has passed since she took possession of the dowry?
He replied: Praise be to Allaah, there are many opinions concerning this matter among the scholars. … The most correct opinion is the view of those who say that no zakaah is due at all until one year has passed, or zakaah is due once when she takes possession of it. There is evidence for both views. The former is the view of Abu Haneefah and the latter is the view of Maalik, and both are mentioned in the madhhab of Ahmad. And Allaah knows best. End quote from Majmoo’ al-Fataawa (25/47). (Source islamqa #93780 -read the whole fatwa for details-)
Note if your husband was not wealthy and couldn't or refused to pay the mahr or the rest of it, it is kept as debt he would have to pay and you only would have to pay your zakat when possessing it.