This may be a silly question, but what if you set p=1 in the binomial distribution probability mass function, doesn't the PMF go to zero? Does this mean the probability of k successes in n trials is zero despite the probability of success in one trial is 100%?
There's 2 cases to look at here.
If $k < n$, we have that the probability of $k$ successes in $n$ trials will be $0$. This is because the binomial distribution parameter, $p$ refers to the probability of success for each individual trial. In the case that $p = 1$, we have that every single trial will be a success. This means that in $n$ trials we will have $n$ successes and so $k$ being less than $n$ guarantees that the probability of having $k$ successes in $n$ trials is $0$.
On the other hand, if $k = n$, using the logic from above, we can find that the probability of $k$ successes in $n$ trials is 1.
Therefore, the PMF is not $0$, as the entire mass of the function is contained at $k = n$.
However, as stated in Rohan's answer, having $p = 1$ doesn't really agree with the idea of a binomial distribution.
Note that a binomial distribution has two possible outcomes: success or failure. As a result, the random variable $X$ in a binomial distribution denotes the no. of successes.
If the probability of success, $p=1$, then it is no longer a binomial distribution!!