Can banks 'create' money on their own or do they need help from other banks?

The question in the title is actually a bit broader than the question in the body, I will try to answer both questions starting with the one in the title.

Answer to Question in the Title

Can banks 'create' money on their own or do they need help from other banks?

Yes, banks actually can crate deposits on their own without help of other banks, thanks to getting excess reserves in the interbank market or from central bank. For example, you can have look at McLeay (2014) Money creation in the modern economy which explains that in greater detail, and it will likely be covered in later chapters in the textbook unless it is old one (or you can have look at other textbooks such as Blanchard et al Macroeconomics a European Perspective).

Answer to the Question in the Body

In the body your question is about whether single bank can increase money supply within the money multiplier model. There the answer is no.

The multiplier model is one model of money creation that applies under certain circumstances, such as when central banks decide to exogenously control money supply by changing quantity of reserves (although the money multiplier exists in multiple models usually when people talk about money multiplier they talk about the simplest exogenous supply of money model). But the multiplier model won't hold (save for special cases) when the central bank conducts its policy through setting interest rate and supplying amount of reserves that is demanded.

This is because if there is only single bank it won't be able to create more reserves out of new deposits For example, let us assume that single bank starts with 100 reserves (R), which are assets  and 100 deposits (D) which are liabilities. The ballance sheet of such bank will look like:

  Bank A
   A | L
 -----------
100R | 100D

Now if bank issues a loan that means that it is converting some of the reserves to a loan. If a single bank would issue a loan and the person would kept the loan with the same bank then the balance sheet would look like this:

  Bank A
   A | L
 -----------
 10R | 100D
 90L |

And bank would not be able to lent any more money as with 10% reserve ratio it would already issue the maximum amount of loans it can make as it has no more excess reserves to lent out. However, the 90 loan that exists the bank can then become a reserve for a new bank, so for bank B the balance sheet would look like:

   Bank B
   A | L
 -----------
 90R | 90D

Hence now that original 90 loan became new 90 reserve and that would not happen if the original bank would just keep the money. Now the new bank can further break the 90R into 10R and 80L and so on.

Now as mentioned in the first paragraph the above won't hold in practice as even single bank nowadays can support more lending by getting excess reserves from other banks  or when central bank is willing to supply additional reserves instead of just setting the quantity reserves from the get go. Nowadays central banks mainly change the price for reserves (i.e. the interest) not amount of reserves. In that case that single bank can just increase the amount of R to create new loans (and matching deposits). However, even that is currently no longer the case because reserve requirements were dropped by Fed in 2020 so now banks can lend as much as they can (at least in US and other countries).

The multiplier model is a simplification that is mostly used as an introduction to money creation since it is an exogenous money supply model and those are easier to solve and learning the more nuanced endogenous money supply model is easier after learning the exogenous one. I am not sure to which Mishkin textbook you refer to, but I would be surprised if in later chapters it would not cover endogenous money supply models (money multiplier is an exogenous money supply model). For example, textbook by Blanchard et al. covers endogenous money supply model right after money multiplier, unless the Mishkin textbook you refer to is very old (prior 2009) it will likely cover it as well in subsequent section.

The key to answering your question is to consider whether or not the bank that has just extended a new loan must make an interbank payment of reserves.

So let bank A extend a loan 100 to customer 1. The bank debits loan assets for an increase 100 and credits deposit liabilities due to customer 1 for an increase 100. The bank balance sheet expands.

Assume customer 1 spends loan funds to customer 2 who also has an account at bank A. Then bank A debits customer 1 deposits for a decrease -100 and credits customer 2 deposits for an increase +100. There is no transfer of bank reserves in the interbank payment system.

If bank A is the only bank in a region it has 100% of the customers. Then bank A offers payment and overdraft (lending) services to the whole money region. These are the same functions offered by the central bank and banking sector in a money union except individual banks must also make interbank transfers of exchange settlement funds or bank reserves.

So if bank A must make a transfer of reserves to bank B to clear payment on a loan 100 then bank A would credit reserves for a decrease -100 and debit deposits due to the initial borrower for a decrease -100. That means bank A must have excess reserves 100 available to clear interbank payment when deposits created by the loan transfer to a customer of bank B. The sources of excess reserves are selling assets out of the liquid securities of the bank or increasing the liabilities and/or equity of the bank on the liabilities and equity side of the balance sheet. So we see that banks issue liabilities and equity to keep reserve payments flowing in the interbank payment system.

This 10 page paper is the best reference I have ever found describing how banks actually operate in realistic network banking systems:

https://think.ing.com/uploads/reports/Money_paradox2.pdf

The money multiplier model assumes there is a reserve limit on the expansion of bank credit (assets) and deposits plus equity (liabilities) in the banking sector. However real banks engage in network banking relations with managers who seek to attract deposits and equity and other managers who underwrite loans by evaluating the credit of prospective borrowers. Real central banks provide payment and overdraft services to the bank sector in addition to other functions.

The quality of loans in the loan portfolio and the size of the equity position available to absorb a loan loss determines whether or not a bank or banking sector is solvent. The ability to rollover liabilities and equity without liquidating portfolio assets determines whether a bank or banking sector is liquid.