Naive version, pops in O(N) if min is popped. Corner cases: you cannot pop an empty stack; and when you pop the last element, min must be hardset to None.

class Node(object):
    def __init__(self, value):
        self.value = value
        self.next = None


class Stack(object):
    top = None
    min = None

    def push(self, value):
        new_node = Node(value)

        if self.top is None:
            self.top = new_node
            self.min = self.top.value
        else:
            new_node.next = self.top
            self.top = new_node

            if self.top.value < self.min:
                self.min = self.top.value


    def pop(self):
        # popping an empty stack:
        if self.top is None:
            print 'cannot pop an empty stack'
        # popping the last element:
        elif self.top.next is None:
            self.top = None
            self.max = None
        else:
            # overwrite max
            if self.max == self.top.value:
                n = self.top.next
                self.max = n.value
                while n is not None:
                    if n.value > self.max:
                        self.max = n.value
                    n = n.next

            self.top = self.top.next
          

    def minim(self):
        print 'min is ' + str(self.min)

    def show(self):
        n = self.top
        while n is not None:
            print n.value,
            n = n.next
        print


s = Stack()
s.push(10)
s.push(20)
s.push(5)
s.push(5)
s.show()
s.minim()
s.pop()
s.minim()
s.pop()
s.minim()

Better version using a second stack, runs in O(1)

class Stack(object):
    def __init__(self):
        self.top = None
        self.max = []

    def push(self, value):
        new_node = Node(value)
        new_node.next = self.top
        self.top = new_node

        if len(self.max) == 0:
            self.max.append(self.top.value)
        else:
            if new_node.value > self.max[-1]:
                self.max.append(new_node.value)
    
    def pop(self):
        if self.top is None:
            return
        else:
            if self.top.value == self.max[-1]:
                self.max.pop()
            self.top = self.top.next