Not specifically, but some lenders will have a limit of one or two mortgages. Others have no caps at all, provided the rental income covers the payments.
Many lenders regard multiple buy to let mortgages in the same area as a higher risk. That is because were rental prices to drop in that area; it could instantly affect your ability to keep up with the repayments on each mortgage.
Lenders will need to see minimum rental income coverage to see that the income will comfortably cover the BTL mortgage interest.
That is usually 125% for lower tax bands and up to 150% if you are a higher-rate taxpayer.
For example - say you want to borrow £50,000 on a BTL mortgage; you might get a very different response from two different lenders:
Lender A requires rental income to cover 150% of the mortgage interest, calculated at 4%. They need to see a rental income of £250 per month to approve the mortgage, with the monthly interest costing £167.
Lender B applies a 6% nominal interest rate and needs to see a rental income of 125% of the monthly mortgage interest. They will approve the application as long as the monthly rent is £313 or more, against a monthly interest charge of £250.
As you can see, a lot depends on the lender's policies and the nominal interest rate they use in their affordability calculations.
Most landlords won't need to prove any other income streams to qualify. Still, new landlords can face additional requirements - in which case an independent broker can help identify the most suitable mortgage lenders to apply to.