Is it possible that the value of a continuous mother wavelet at origin is zero, i.e. $\psi(t=0)=0$?
Yes: a sufficient criterion for a valid wavelet is being zero mean in time domain, i.e. $\hat \psi (0) = 0$. For $\psi(0) = 0$ we require that $\hat \psi$ sums to zero; one example is higher order Generalized Morse Wavelets - from Olhede &amp; Walden,

(Though above is only approximately zero-sum). Such wavelets can be admissible and analytic, enabling CWT inversion. Trivial Python example below; GMWs are also implemented in ssqueezepy.

Code
import numpy as np
import matplotlib.pyplot as plt
from numpy.fft import ifft, ifftshift

t = np.linspace(0, 1, 256, endpoint=False)
wf = np.exp(-(t - .08)**2 * 4096)
wf -= np.roll(wf, 25)    # shift by 25 samples and subtract
w = ifftshift(ifft(wf))  # take to time domain and center

plt.plot(wf)
plt.show()

plt.plot(w.real)
plt.plot(w.imag)
plt.show()