Baby Shower & no extra guests please

What is a nice way of saying no plus ones allowed on the invitations? I mentioned the shower to one friend & said him and his wife would be invited...his response the next day was that the WHOLE family would be coming. How does an invite for 2 turn into a party of 8??!!!! Another person my mom’s friend said she wanted to bring their mother & daughter with them (We have never met neither her mother or child before). I’m SO confused why people think this is acceptable & assume they can bring whoever.

We cannot afford the extra people who are not invited & the venue space is limited. So we plan on having people RSVP but I’m still afraid other people will take it upon themselves to invite others with them & have this same mindset.

So how do we word the invite nicely and to where it is clear that we can’t take on extras?