There is a basic rule saying that during the divorce proceedings the foreigner living on a temporary residence permit as a spouse of a Pole can still legally reside in Poland. Only after the divorce proceedings are over and the divorce ruling becomes final, the current temporary residence permit of the foreigner is to be cancelled. In this case the foreigner has 15 days to inform the competent authority (Urząd) about the fact of the divorce. Then the authority should take a decision on the cancellation of the temporary residence permit. However, till the decision is not taken, the divorced person may submit a request for a temporary residence permit on another ground. Apart from classical grounds such us working purposes, studies or business activities, one should take into account a chance to get residence permit after divorce or separation on the basis of a “vital interest”. This possibility is directly mentioned in article 158 para. 2(1) of the Polish Law on Foreigners.
The most usual example of such a vital interest is the need to exercise parental authority over the child. It means that a foreign parent may legally prolong his or her stay in Poland, if the child is in his or her custody as the result of divorce or legal separation. Therefore the person wishing to obtain resident permit on this ground shall submit to the competent body the divorce ruling or the document from the register proving that the divorce took place. Besides, either divorce ruling itself or any additional documents handed down to the competent body shall contain a proof of a vital interest for remaining in Poland. It might be, for example, the court judgement which clearly indicates that the common child is to reside together with the former spouse applying to the Urząd. Apart from the mentioned documents, you still need to submit a classical package including a filled-in application form with photos, an insurance contract, the birth certificate of your child, a tenancy contract (showing that you have an accommodation) and a document proving your stable and regular income (alimonies fixed by the court judgment might be the source of your income)
However, one should take into consideration that the term of the temporary residence permit on the described ground is strictly limited. It can be issued for no longer than 3 years. Moreover, after divorce it becomes a bit harder to get the permanent residence permit and, consequently, the citizenship. Nevertheless, one can apply for a long-term EU residence permit after 5 years of living in Poland. In turn, after 3 years of possessing such a long-term residence permit, one is entitled to apply for Polish citizenship.
Summing up, it seems plausible to admit that a person taking care of his or her Polish child is able to prolong the residence permit even after divorce with the Polish spouse. However, in this case one should take into account the obligation to inform the competent authority about the divorce and submit a new application without any postponement. Without any doubt, it would be reasonable to consult a professional lawyer in such a case to be sure that all the formal requirements are fulfilled.