An experiment to search for the $K^-pp$ bound state was performed via the in-flight $^3$He($K^-,n)$ reaction using 5.3 $times$ $10^9$ kaons at 1 GeV/$c$ at the J-PARC hadron experimental facility. In the semi-inclusive neutron missing-mass spectrum a
t $theta_{n}^{lab}=0^circ$, no significant peak was observed in the region corresponding to $K^-pp$ binding energy larger than 80 MeV, where a bump structure has been reported in the $Lambda p$ final state in different reactions. Assuming the state to be isotropically decaying into $Lambda p$, mass-dependent upper limits on the production cross section were determined to be 30--180, 70--250, and 100--270 $mu$b/sr, for the natural widths of 20, 60, and 100 MeV, respectively, at 95% confidence level.
