That was my first thought.
The "mutual" friend says its not much of an ask and you should do it.. even though you are reluctant? Because she thinks if you don't she'll have to do it. Also.. the mutual friend going on about it takes a village - sounds like she's been thinking of asking you for favours too. I don't think she's a friend and I also think she will tell the CF, since she sees so much of the virtue in CF's request to you.
Dear @Addiee5 PLEASE PLEASE PLEASE... repeat this to yourself whenever you doubt yourself. You do NOT need permission from us, or from any mutual friend to say NO to this woman asking you to do something which would be awkward and difficult for you to do.
You just don't have to say yes. It's not mean. Its prioritising yourself and your child. It doesn't make you a bad person not to volunteer to make someone else's life run more smoothly. That is her job. Not yours.
Part of the relucance to say no, can be that you have been conditioned to be supportive and helpful in childhood but there's a fear that if you say no, there will be recriminations, other people might think that you are not a nice person. If she is the sort of woman to run around school bitching that you let her down...most people will be looking at her behaviour, not yours... It might even stop others from deciding that you can help them out too.
You just don't have to say yes. It's not mean. Its prioritising yourself and your child. It doesn't make you a bad person not to volunteer to make someone else's life run more smoothly. That is her job. Not yours. If she wants a regular pick up she can advertise and pay. I'm not talking about playdate pick up. She's also not a particular friend of yours and you owe her nothing.
I don't believe that other mums sit around saying.. "ooh Mrs Y was asked to pick up Mrs X's DD every day and she said no.. how mean is that?" Every single mum at the school gate knows that regularly picking up other people's children, especially if it goes on for a while is essentially blagging free childcare on the promise that occasionally they will return the favour.
I know how you feel. I've been in your position and in the first school my DC's attended people did help each other out. In the second one.. I was often asked and promised, but when we had a serious accident to deal with, not one of the many people I did (considerable) favours for offered to help in any way at all. When they rang, it was for gossip purposes. It was eye opening. Surprisingly, it was other (kinder) people who offered support and I will never forget that.
I'm not sure how old your DD or what school year, but you will make friends eventually but make the right ones. Just focus on playdates and socialising for your DD.. do plenty of non school activities with her so everything is not all about school - for her and for you.
You will not be tarred and feathered for saying no. If its difficult face to face. Text it and have some good phrases up your sleeve for a face to face. The best replies were along the lines of I can't guarantee and I don't know in advance. So I have to say no. or like @Goditsmemargaret s post
Best of luck xx